Step |
Hyp |
Ref |
Expression |
1 |
|
imaeq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝐹 “ 𝑦 ) = ( 𝐹 “ 𝐴 ) ) |
2 |
1
|
sseq2d |
⊢ ( 𝑦 = 𝐴 → ( 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ↔ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) ) |
3 |
2
|
anbi2d |
⊢ ( 𝑦 = 𝐴 → ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) ↔ ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) ) ) |
4 |
|
sseq2 |
⊢ ( 𝑦 = 𝐴 → ( 𝑥 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝐴 ) ) |
5 |
4
|
anbi1d |
⊢ ( 𝑦 = 𝐴 → ( ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ↔ ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ) |
6 |
5
|
exbidv |
⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ↔ ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ) |
7 |
3 6
|
imbi12d |
⊢ ( 𝑦 = 𝐴 → ( ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ↔ ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ) ) |
8 |
|
vex |
⊢ 𝑦 ∈ V |
9 |
8
|
ssimaex |
⊢ ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝑦 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝑦 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) |
10 |
7 9
|
vtoclg |
⊢ ( 𝐴 ∈ 𝐶 → ( ( Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) ) |
11 |
10
|
3impib |
⊢ ( ( 𝐴 ∈ 𝐶 ∧ Fun 𝐹 ∧ 𝐵 ⊆ ( 𝐹 “ 𝐴 ) ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐴 ∧ 𝐵 = ( 𝐹 “ 𝑥 ) ) ) |