Step |
Hyp |
Ref |
Expression |
1 |
|
axdc2.1 |
⊢ 𝐴 ∈ V |
2 |
|
eleq1w |
⊢ ( 𝑠 = 𝑥 → ( 𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 ) ) |
3 |
2
|
adantr |
⊢ ( ( 𝑠 = 𝑥 ∧ 𝑡 = 𝑦 ) → ( 𝑠 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 ) ) |
4 |
|
fveq2 |
⊢ ( 𝑠 = 𝑥 → ( 𝐹 ‘ 𝑠 ) = ( 𝐹 ‘ 𝑥 ) ) |
5 |
4
|
eleq2d |
⊢ ( 𝑠 = 𝑥 → ( 𝑡 ∈ ( 𝐹 ‘ 𝑠 ) ↔ 𝑡 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
6 |
|
eleq1w |
⊢ ( 𝑡 = 𝑦 → ( 𝑡 ∈ ( 𝐹 ‘ 𝑥 ) ↔ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
7 |
5 6
|
sylan9bb |
⊢ ( ( 𝑠 = 𝑥 ∧ 𝑡 = 𝑦 ) → ( 𝑡 ∈ ( 𝐹 ‘ 𝑠 ) ↔ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
8 |
3 7
|
anbi12d |
⊢ ( ( 𝑠 = 𝑥 ∧ 𝑡 = 𝑦 ) → ( ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ( 𝐹 ‘ 𝑠 ) ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) ) ) |
9 |
8
|
cbvopabv |
⊢ { 〈 𝑠 , 𝑡 〉 ∣ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ( 𝐹 ‘ 𝑠 ) ) } = { 〈 𝑥 , 𝑦 〉 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑥 ) ) } |
10 |
|
fveq2 |
⊢ ( 𝑛 = 𝑥 → ( ℎ ‘ 𝑛 ) = ( ℎ ‘ 𝑥 ) ) |
11 |
10
|
cbvmptv |
⊢ ( 𝑛 ∈ ω ↦ ( ℎ ‘ 𝑛 ) ) = ( 𝑥 ∈ ω ↦ ( ℎ ‘ 𝑥 ) ) |
12 |
1 9 11
|
axdc2lem |
⊢ ( ( 𝐴 ≠ ∅ ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝐴 ∖ { ∅ } ) ) → ∃ 𝑔 ( 𝑔 : ω ⟶ 𝐴 ∧ ∀ 𝑘 ∈ ω ( 𝑔 ‘ suc 𝑘 ) ∈ ( 𝐹 ‘ ( 𝑔 ‘ 𝑘 ) ) ) ) |