Step |
Hyp |
Ref |
Expression |
1 |
|
df-f |
⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ) |
2 |
1
|
a1i |
⊢ ( 𝑋 ∈ 𝑉 → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ) ) |
3 |
2
|
sbcbidv |
⊢ ( 𝑋 ∈ 𝑉 → ( [ 𝑋 / 𝑥 ] 𝐹 : 𝐴 ⟶ 𝐵 ↔ [ 𝑋 / 𝑥 ] ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ) ) |
4 |
|
sbcfng |
⊢ ( 𝑋 ∈ 𝑉 → ( [ 𝑋 / 𝑥 ] 𝐹 Fn 𝐴 ↔ ⦋ 𝑋 / 𝑥 ⦌ 𝐹 Fn ⦋ 𝑋 / 𝑥 ⦌ 𝐴 ) ) |
5 |
|
sbcssg |
⊢ ( 𝑋 ∈ 𝑉 → ( [ 𝑋 / 𝑥 ] ran 𝐹 ⊆ 𝐵 ↔ ⦋ 𝑋 / 𝑥 ⦌ ran 𝐹 ⊆ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ) ) |
6 |
|
csbrn |
⊢ ⦋ 𝑋 / 𝑥 ⦌ ran 𝐹 = ran ⦋ 𝑋 / 𝑥 ⦌ 𝐹 |
7 |
6
|
sseq1i |
⊢ ( ⦋ 𝑋 / 𝑥 ⦌ ran 𝐹 ⊆ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ↔ ran ⦋ 𝑋 / 𝑥 ⦌ 𝐹 ⊆ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ) |
8 |
5 7
|
bitrdi |
⊢ ( 𝑋 ∈ 𝑉 → ( [ 𝑋 / 𝑥 ] ran 𝐹 ⊆ 𝐵 ↔ ran ⦋ 𝑋 / 𝑥 ⦌ 𝐹 ⊆ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ) ) |
9 |
4 8
|
anbi12d |
⊢ ( 𝑋 ∈ 𝑉 → ( ( [ 𝑋 / 𝑥 ] 𝐹 Fn 𝐴 ∧ [ 𝑋 / 𝑥 ] ran 𝐹 ⊆ 𝐵 ) ↔ ( ⦋ 𝑋 / 𝑥 ⦌ 𝐹 Fn ⦋ 𝑋 / 𝑥 ⦌ 𝐴 ∧ ran ⦋ 𝑋 / 𝑥 ⦌ 𝐹 ⊆ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ) ) ) |
10 |
|
sbcan |
⊢ ( [ 𝑋 / 𝑥 ] ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ↔ ( [ 𝑋 / 𝑥 ] 𝐹 Fn 𝐴 ∧ [ 𝑋 / 𝑥 ] ran 𝐹 ⊆ 𝐵 ) ) |
11 |
|
df-f |
⊢ ( ⦋ 𝑋 / 𝑥 ⦌ 𝐹 : ⦋ 𝑋 / 𝑥 ⦌ 𝐴 ⟶ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ↔ ( ⦋ 𝑋 / 𝑥 ⦌ 𝐹 Fn ⦋ 𝑋 / 𝑥 ⦌ 𝐴 ∧ ran ⦋ 𝑋 / 𝑥 ⦌ 𝐹 ⊆ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ) ) |
12 |
9 10 11
|
3bitr4g |
⊢ ( 𝑋 ∈ 𝑉 → ( [ 𝑋 / 𝑥 ] ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ↔ ⦋ 𝑋 / 𝑥 ⦌ 𝐹 : ⦋ 𝑋 / 𝑥 ⦌ 𝐴 ⟶ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ) ) |
13 |
3 12
|
bitrd |
⊢ ( 𝑋 ∈ 𝑉 → ( [ 𝑋 / 𝑥 ] 𝐹 : 𝐴 ⟶ 𝐵 ↔ ⦋ 𝑋 / 𝑥 ⦌ 𝐹 : ⦋ 𝑋 / 𝑥 ⦌ 𝐴 ⟶ ⦋ 𝑋 / 𝑥 ⦌ 𝐵 ) ) |