Step |
Hyp |
Ref |
Expression |
1 |
|
rnmpt.1 |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
2 |
|
vex |
⊢ 𝑥 ∈ V |
3 |
|
id |
⊢ ( 𝑥 = 𝑧 → 𝑥 = 𝑧 ) |
4 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑧 → 𝐴 = ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ) |
5 |
3 4
|
eleq12d |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐴 ↔ 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ) ) |
6 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑧 → 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
7 |
6
|
biantrud |
⊢ ( 𝑥 = 𝑧 → ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ↔ ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ) |
8 |
5 7
|
bitr2d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ↔ 𝑥 ∈ 𝐴 ) ) |
9 |
8
|
equcoms |
⊢ ( 𝑧 = 𝑥 → ( ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ↔ 𝑥 ∈ 𝐴 ) ) |
10 |
2 9
|
spcev |
⊢ ( 𝑥 ∈ 𝐴 → ∃ 𝑧 ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
11 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ) |
12 |
|
nfv |
⊢ Ⅎ 𝑧 ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) |
13 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑧 / 𝑥 ⦌ 𝐴 |
14 |
13
|
nfcri |
⊢ Ⅎ 𝑥 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 |
15 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
16 |
15
|
nfeq2 |
⊢ Ⅎ 𝑥 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
17 |
14 16
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
18 |
6
|
eqeq2d |
⊢ ( 𝑥 = 𝑧 → ( 𝑦 = 𝐵 ↔ 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
19 |
5 18
|
anbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ) |
20 |
12 17 19
|
cbvexv1 |
⊢ ( ∃ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑧 ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
21 |
11 20
|
bitri |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃ 𝑧 ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
22 |
|
eqeq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ↔ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
23 |
22
|
anbi2d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ↔ ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ) |
24 |
23
|
exbidv |
⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑧 ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝑦 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ↔ ∃ 𝑧 ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ) |
25 |
21 24
|
syl5bb |
⊢ ( 𝑦 = 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃ 𝑧 ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ) |
26 |
1
|
rnmpt |
⊢ ran 𝐹 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 } |
27 |
25 26
|
elab2g |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ∈ ran 𝐹 ↔ ∃ 𝑧 ( 𝑧 ∈ ⦋ 𝑧 / 𝑥 ⦌ 𝐴 ∧ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ) |
28 |
10 27
|
syl5ibr |
⊢ ( 𝐵 ∈ 𝑉 → ( 𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹 ) ) |
29 |
28
|
impcom |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉 ) → 𝐵 ∈ ran 𝐹 ) |