Step |
Hyp |
Ref |
Expression |
1 |
|
rexima.x |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑦 ) → ( 𝜑 ↔ 𝜓 ) ) |
2 |
|
fvexd |
⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑦 ) ∈ V ) |
3 |
|
fvelimab |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑥 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐵 ( 𝐹 ‘ 𝑦 ) = 𝑥 ) ) |
4 |
|
eqcom |
⊢ ( ( 𝐹 ‘ 𝑦 ) = 𝑥 ↔ 𝑥 = ( 𝐹 ‘ 𝑦 ) ) |
5 |
4
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐵 ( 𝐹 ‘ 𝑦 ) = 𝑥 ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = ( 𝐹 ‘ 𝑦 ) ) |
6 |
3 5
|
bitrdi |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑥 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑦 ∈ 𝐵 𝑥 = ( 𝐹 ‘ 𝑦 ) ) ) |
7 |
1
|
adantl |
⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) ∧ 𝑥 = ( 𝐹 ‘ 𝑦 ) ) → ( 𝜑 ↔ 𝜓 ) ) |
8 |
2 6 7
|
ralxfr2d |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ∀ 𝑥 ∈ ( 𝐹 “ 𝐵 ) 𝜑 ↔ ∀ 𝑦 ∈ 𝐵 𝜓 ) ) |