α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
⊢ (∃ p, (#↑p) = c) ↔ ∃ s_1, (#↑(Subtype.val '' s_1)) = c
α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
⊢ c ≤ (#↑s) ↔ ∃ p, p ⊆ s ∧ (#↑p) = c
rw [le_mk_iff_exists_set, ← Subtype.exists_set_subtype]
case h
α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
⊢ ∀ (a : Set ↑s), (#↑a) = c ↔ (#↑(Subtype.val '' a)) = c
α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
⊢ (∃ p, (#↑p) = c) ↔ ∃ s_1, (#↑(Subtype.val '' s_1)) = c
apply exists_congr
case h
α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
t : Set ↑s
⊢ (#↑t) = c ↔ (#↑(Subtype.val '' t)) = c
case h
α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
⊢ ∀ (a : Set ↑s), (#↑a) = c ↔ (#↑(Subtype.val '' a)) = c
intro t
case h
α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
t : Set ↑s
⊢ Injective Subtype.val
case h
α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
t : Set ↑s
⊢ (#↑t) = c ↔ (#↑(Subtype.val '' t)) = c
rw [mk_image_eq]
no goals
case h
α✝ β : Type u
c : Cardinal
α : Type u
s : Set α
t : Set ↑s
⊢ Injective Subtype.val
apply Subtype.val_injective