Step |
Hyp |
Ref |
Expression |
1 |
|
pweq |
⊢ ( 𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋 ) |
2 |
1
|
difeq1d |
⊢ ( 𝑦 = 𝑋 → ( 𝒫 𝑦 ∖ { ∅ } ) = ( 𝒫 𝑋 ∖ { ∅ } ) ) |
3 |
2
|
oveq1d |
⊢ ( 𝑦 = 𝑋 → ( ( 𝒫 𝑦 ∖ { ∅ } ) ↑m 𝐴 ) = ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ) |
4 |
3
|
raleqdv |
⊢ ( 𝑦 = 𝑋 → ( ∀ 𝑓 ∈ ( ( 𝒫 𝑦 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |
5 |
4
|
anbi2d |
⊢ ( 𝑦 = 𝑋 → ( ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑦 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ↔ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) ) |
6 |
|
df-acn |
⊢ AC 𝐴 = { 𝑦 ∣ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑦 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) } |
7 |
5 6
|
elab2g |
⊢ ( 𝑋 ∈ 𝑉 → ( 𝑋 ∈ AC 𝐴 ↔ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) ) |
8 |
|
elex |
⊢ ( 𝐴 ∈ 𝑊 → 𝐴 ∈ V ) |
9 |
|
biid |
⊢ ( ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ↔ ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |
10 |
9
|
baib |
⊢ ( 𝐴 ∈ V → ( ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |
11 |
8 10
|
syl |
⊢ ( 𝐴 ∈ 𝑊 → ( ( 𝐴 ∈ V ∧ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |
12 |
7 11
|
sylan9bb |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ) → ( 𝑋 ∈ AC 𝐴 ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |