Step |
Hyp |
Ref |
Expression |
1 |
|
abfmpeld.1 |
⊢ 𝐹 = ( 𝑥 ∈ 𝑉 ↦ { 𝑦 ∣ 𝜓 } ) |
2 |
|
abfmpeld.2 |
⊢ ( 𝜑 → { 𝑦 ∣ 𝜓 } ∈ V ) |
3 |
|
abfmpeld.3 |
⊢ ( 𝜑 → ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝜓 ↔ 𝜒 ) ) ) |
4 |
2
|
alrimiv |
⊢ ( 𝜑 → ∀ 𝑥 { 𝑦 ∣ 𝜓 } ∈ V ) |
5 |
|
csbexg |
⊢ ( ∀ 𝑥 { 𝑦 ∣ 𝜓 } ∈ V → ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } ∈ V ) |
6 |
4 5
|
syl |
⊢ ( 𝜑 → ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } ∈ V ) |
7 |
1
|
fvmpts |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } ∈ V ) → ( 𝐹 ‘ 𝐴 ) = ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } ) |
8 |
6 7
|
sylan2 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( 𝐹 ‘ 𝐴 ) = ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } ) |
9 |
|
csbab |
⊢ ⦋ 𝐴 / 𝑥 ⦌ { 𝑦 ∣ 𝜓 } = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } |
10 |
8 9
|
eqtrdi |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( 𝐹 ‘ 𝐴 ) = { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ) |
11 |
10
|
eleq2d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ) ) |
12 |
11
|
adantl |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ) ) |
13 |
|
simpll |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ∧ 𝑦 = 𝐵 ) → 𝐴 ∈ 𝑉 ) |
14 |
3
|
ancomsd |
⊢ ( 𝜑 → ( ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) ) |
15 |
14
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( ( 𝑦 = 𝐵 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) ) |
16 |
15
|
impl |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ∧ 𝑦 = 𝐵 ) ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) ) |
17 |
13 16
|
sbcied |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ∧ 𝑦 = 𝐵 ) → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) |
18 |
17
|
ex |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) ) |
19 |
18
|
alrimiv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) → ∀ 𝑦 ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) ) |
20 |
|
elabgt |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ ∀ 𝑦 ( 𝑦 = 𝐵 → ( [ 𝐴 / 𝑥 ] 𝜓 ↔ 𝜒 ) ) ) → ( 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ↔ 𝜒 ) ) |
21 |
19 20
|
sylan2 |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ) → ( 𝐵 ∈ { 𝑦 ∣ [ 𝐴 / 𝑥 ] 𝜓 } ↔ 𝜒 ) ) |
22 |
12 21
|
bitrd |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝜑 ) ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝜒 ) ) |
23 |
22
|
an13s |
⊢ ( ( 𝜑 ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝜒 ) ) |
24 |
23
|
ex |
⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ∈ ( 𝐹 ‘ 𝐴 ) ↔ 𝜒 ) ) ) |