Step |
Hyp |
Ref |
Expression |
1 |
|
fvssunirn |
⊢ ( 𝑓 ‘ 𝑥 ) ⊆ ∪ ran 𝑓 |
2 |
|
simpr |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) ) → 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) ) |
3 |
1 2
|
sselid |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) ) → 𝐵 ∈ ∪ ran 𝑓 ) |
4 |
3
|
ralimiaa |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓 ) |
5 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
6 |
5
|
fmpt |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ∪ ran 𝑓 ↔ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ∪ ran 𝑓 ) |
7 |
4 6
|
sylib |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ∪ ran 𝑓 ) |
8 |
|
id |
⊢ ( 𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉 ) |
9 |
|
vex |
⊢ 𝑓 ∈ V |
10 |
9
|
rnex |
⊢ ran 𝑓 ∈ V |
11 |
10
|
uniex |
⊢ ∪ ran 𝑓 ∈ V |
12 |
|
fex2 |
⊢ ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉 ∧ ∪ ran 𝑓 ∈ V ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ V ) |
13 |
11 12
|
mp3an3 |
⊢ ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ∪ ran 𝑓 ∧ 𝐴 ∈ 𝑉 ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ V ) |
14 |
7 8 13
|
syl2anr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ V ) |
15 |
5
|
fvmpt2 |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) = 𝐵 ) |
16 |
15 2
|
eqeltrd |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) ) → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) |
17 |
16
|
ralimiaa |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) → ∀ 𝑥 ∈ 𝐴 ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) ) → ∀ 𝑥 ∈ 𝐴 ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) |
19 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
20 |
19
|
nfeq2 |
⊢ Ⅎ 𝑥 𝑔 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
21 |
|
fveq1 |
⊢ ( 𝑔 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( 𝑔 ‘ 𝑥 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ) |
22 |
21
|
eleq1d |
⊢ ( 𝑔 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |
23 |
20 22
|
ralbid |
⊢ ( 𝑔 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) ) |
24 |
14 18 23
|
spcedv |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ( 𝑓 ‘ 𝑥 ) ) → ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) |