Step |
Hyp |
Ref |
Expression |
1 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
2 |
1
|
eleq2d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
3 |
2
|
ralbidv |
⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
4 |
3
|
exbidv |
⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
5 |
|
acnrcl |
⊢ ( 𝑋 ∈ AC 𝐴 → 𝐴 ∈ V ) |
6 |
|
isacn |
⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐴 ∈ V ) → ( 𝑋 ∈ AC 𝐴 ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |
7 |
5 6
|
mpdan |
⊢ ( 𝑋 ∈ AC 𝐴 → ( 𝑋 ∈ AC 𝐴 ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |
8 |
7
|
ibi |
⊢ ( 𝑋 ∈ AC 𝐴 → ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) |
9 |
8
|
adantr |
⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) |
10 |
|
pwexg |
⊢ ( 𝑋 ∈ AC 𝐴 → 𝒫 𝑋 ∈ V ) |
11 |
10
|
difexd |
⊢ ( 𝑋 ∈ AC 𝐴 → ( 𝒫 𝑋 ∖ { ∅ } ) ∈ V ) |
12 |
11 5
|
elmapd |
⊢ ( 𝑋 ∈ AC 𝐴 → ( 𝐹 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) ) |
13 |
12
|
biimpar |
⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → 𝐹 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ) |
14 |
4 9 13
|
rspcdva |
⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) |