Step |
Hyp |
Ref |
Expression |
1 |
|
ralima.x |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑦 ) → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
fnfun |
⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 ) |
3 |
2
|
funfnd |
⊢ ( 𝐹 Fn 𝐴 → 𝐹 Fn dom 𝐹 ) |
4 |
|
fndm |
⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 ) |
5 |
4
|
sseq2d |
⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴 ) ) |
6 |
5
|
biimpar |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ dom 𝐹 ) |
7 |
|
fvexd |
⊢ ( ( ( 𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑦 ) ∈ V ) |
8 |
|
fvelimab |
⊢ ( ( 𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹 ) → ( 𝑥 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐵 ( 𝐹 ‘ 𝑦 ) = 𝑥 ) ) |
9 |
|
eqcom |
⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑥 ↔ 𝑥 = ( 𝐹 ‘ 𝑦 ) ) |
10 |
9
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐵 ( 𝐹 ‘ 𝑦 ) = 𝑥 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = ( 𝐹 ‘ 𝑦 ) ) |
11 |
8 10
|
bitrdi |
⊢ ( ( 𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹 ) → ( 𝑥 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = ( 𝐹 ‘ 𝑦 ) ) ) |
12 |
1
|
adantl |
⊢ ( ( ( 𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹 ) ∧ 𝑥 = ( 𝐹 ‘ 𝑦 ) ) → ( 𝜑 ↔ 𝜓 ) ) |
13 |
7 11 12
|
ralxfr2d |
⊢ ( ( 𝐹 Fn dom 𝐹 ∧ 𝐵 ⊆ dom 𝐹 ) → ( ∀ 𝑥 ∈ ( 𝐹 “ 𝐵 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) ) |
14 |
3 6 13
|
syl2an2r |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ∀ 𝑥 ∈ ( 𝐹 “ 𝐵 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) ) |