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  Theorem fundmen 6043
   Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by SF, 23-Feb-2015.)
  Hypothesis
  Ref Expression
  fundmen.1 F V
  Assertion
  Ref Expression
  fundmen (Fun F → dom FF)
  
  Proof of Theorem fundmen
  Dummy variables x y z a b are mutually distinct and distinct from all other variables.
  StepHypRef Expression
  1 ssv 3291 . . . . . 6 F V
  2 1stfo 5505 . . . . . . 7 1st :V–onto→V
  3 fofn 5271 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
  4 fnssresb 5195 . . . . . . 7 (1st Fn V → ((1st F) Fn FF V))
  52, 3, 4mp2b 9 . . . . . 6 ((1st F) Fn FF V)
  61, 5mpbir 200 . . . . 5 (1st F) Fn F
  76a1i 10 . . . 4 (Fun F → (1st F) Fn F)
  8 brcnv 4892 . . . . . . . . . . 11 (x(1st F)yy(1st F)x)
  9 brres 4949 . . . . . . . . . . 11 (y(1st F)x ↔ (y1st x y F))
  10 vex 2862 . . . . . . . . . . . . . 14 x V
  1110br1st 4858 . . . . . . . . . . . . 13 (y1st xa y = x, a)
  1211anbi1i 676 . . . . . . . . . . . 12 ((y1st x y F) ↔ (a y = x, a y F))
  13 19.41v 1901 . . . . . . . . . . . 12 (a(y = x, a y F) ↔ (a y = x, a y F))
  1412, 13bitr4i 243 . . . . . . . . . . 11 ((y1st x y F) ↔ a(y = x, a y F))
  158, 9, 143bitri 262 . . . . . . . . . 10 (x(1st F)ya(y = x, a y F))
  16 brcnv 4892 . . . . . . . . . . . 12 (x(1st F)zz(1st F)x)
  17 brres 4949 . . . . . . . . . . . 12 (z(1st F)x ↔ (z1st x z F))
  1810br1st 4858 . . . . . . . . . . . . 13 (z1st xb z = x, b)
  1918anbi1i 676 . . . . . . . . . . . 12 ((z1st x z F) ↔ (b z = x, b z F))
  2016, 17, 193bitri 262 . . . . . . . . . . 11 (x(1st F)z ↔ (b z = x, b z F))
  21 19.41v 1901 . . . . . . . . . . 11 (b(z = x, b z F) ↔ (b z = x, b z F))
  2220, 21bitr4i 243 . . . . . . . . . 10 (x(1st F)zb(z = x, b z F))
  2315, 22anbi12i 678 . . . . . . . . 9 ((x(1st F)y x(1st F)z) ↔ (a(y = x, a y F) b(z = x, b z F)))
  24 eeanv 1913 . . . . . . . . 9 (ab((y = x, a y F) (z = x, b z F)) ↔ (a(y = x, a y F) b(z = x, b z F)))
  2523, 24bitr4i 243 . . . . . . . 8 ((x(1st F)y x(1st F)z) ↔ ab((y = x, a y F) (z = x, b z F)))
  26 an4 797 . . . . . . . . . 10 (((y = x, a y F) (z = x, b z F)) ↔ ((y = x, a z = x, b) (y F z F)))
  27 dffun4 5121 . . . . . . . . . . . . 13 (Fun Fxab((x, a F x, b F) → a = b))
  28 sp 1747 . . . . . . . . . . . . . . 15 (b((x, a F x, b F) → a = b) → ((x, a F x, b F) → a = b))
  2928sps 1754 . . . . . . . . . . . . . 14 (ab((x, a F x, b F) → a = b) → ((x, a F x, b F) → a = b))
  3029sps 1754 . . . . . . . . . . . . 13 (xab((x, a F x, b F) → a = b) → ((x, a F x, b F) → a = b))
  3127, 30sylbi 187 . . . . . . . . . . . 12 (Fun F → ((x, a F x, b F) → a = b))
  32 opeq2 4579 . . . . . . . . . . . 12 (a = bx, a = x, b)
  3331, 32syl6 29 . . . . . . . . . . 11 (Fun F → ((x, a F x, b F) → x, a = x, b))
  34 eleq1 2413 . . . . . . . . . . . . . . 15 (y = x, a → (y Fx, a F))
  35 eleq1 2413 . . . . . . . . . . . . . . 15 (z = x, b → (z Fx, b F))
  3634, 35bi2anan9 843 . . . . . . . . . . . . . 14 ((y = x, a z = x, b) → ((y F z F) ↔ (x, a F x, b F)))
  37 eqeq12 2365 . . . . . . . . . . . . . 14 ((y = x, a z = x, b) → (y = zx, a = x, b))
  3836, 37imbi12d 311 . . . . . . . . . . . . 13 ((y = x, a z = x, b) → (((y F z F) → y = z) ↔ ((x, a F x, b F) → x, a = x, b)))
  3938biimprcd 216 . . . . . . . . . . . 12 (((x, a F x, b F) → x, a = x, b) → ((y = x, a z = x, b) → ((y F z F) → y = z)))
  4039imp3a 420 . . . . . . . . . . 11 (((x, a F x, b F) → x, a = x, b) → (((y = x, a z = x, b) (y F z F)) → y = z))
  4133, 40syl 15 . . . . . . . . . 10 (Fun F → (((y = x, a z = x, b) (y F z F)) → y = z))
  4226, 41syl5bi 208 . . . . . . . . 9 (Fun F → (((y = x, a y F) (z = x, b z F)) → y = z))
  4342exlimdvv 1637 . . . . . . . 8 (Fun F → (ab((y = x, a y F) (z = x, b z F)) → y = z))
  4425, 43syl5bi 208 . . . . . . 7 (Fun F → ((x(1st F)y x(1st F)z) → y = z))
  4544alrimiv 1631 . . . . . 6 (Fun Fz((x(1st F)y x(1st F)z) → y = z))
  4645alrimivv 1632 . . . . 5 (Fun Fxyz((x(1st F)y x(1st F)z) → y = z))
  47 dffun2 5119 . . . . 5 (Fun (1st F) ↔ xyz((x(1st F)y x(1st F)z) → y = z))
  4846, 47sylibr 203 . . . 4 (Fun F → Fun (1st F))
  49 dfdm4 5507 . . . . . 6 dom F = (1stF)
  50 dfima3 4951 . . . . . 6 (1stF) = ran (1st F)
  5149, 50eqtr2i 2374 . . . . 5 ran (1st F) = dom F
  5251a1i 10 . . . 4 (Fun F → ran (1st F) = dom F)
  53 dff1o2 5291 . . . 4 ((1st F):F1-1-onto→dom F ↔ ((1st F) Fn F Fun (1st F) ran (1st F) = dom F))
  547, 48, 52, 53syl3anbrc 1136 . . 3 (Fun F → (1st F):F1-1-onto→dom F)
  55 1stex 4739 . . . . 5 1st V
  56 fundmen.1 . . . . 5 F V
  5755, 56resex 5117 . . . 4 (1st F) V
  5857f1oen 6033 . . 3 ((1st F):F1-1-onto→dom FF ≈ dom F)
  5954, 58syl 15 . 2 (Fun FF ≈ dom F)
  60 ensym 6037 . 2 (F ≈ dom F ↔ dom FF)
  6159, 60sylib 188 1 (Fun F → dom FF)
   Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ⊆ wss 3257  ⟨cop 4561   class class class wbr 4639  1st c1st 4717   “ cima 4722  ◡ccnv 4771  dom cdm 4772  ran crn 4773   ↾ cres 4774  Fun wfun 4775   Fn wfn 4776  –onto→wfo 4779  –1-1-onto→wf1o 4780   ≈ cen 6028 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-en 6029 This theorem is referenced by:  fundmeng  6044  xpsnen  6049
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