Step |
Hyp |
Ref |
Expression |
1 |
|
rclexi.1 |
⊢ 𝐴 ∈ 𝑉 |
2 |
|
ssun1 |
⊢ 𝐴 ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) |
3 |
|
dmun |
⊢ dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) = ( dom 𝐴 ∪ dom ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) |
4 |
|
dmresi |
⊢ dom ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) = ( dom 𝐴 ∪ ran 𝐴 ) |
5 |
4
|
uneq2i |
⊢ ( dom 𝐴 ∪ dom ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) = ( dom 𝐴 ∪ ( dom 𝐴 ∪ ran 𝐴 ) ) |
6 |
|
ssun1 |
⊢ dom 𝐴 ⊆ ( dom 𝐴 ∪ ran 𝐴 ) |
7 |
|
ssequn1 |
⊢ ( dom 𝐴 ⊆ ( dom 𝐴 ∪ ran 𝐴 ) ↔ ( dom 𝐴 ∪ ( dom 𝐴 ∪ ran 𝐴 ) ) = ( dom 𝐴 ∪ ran 𝐴 ) ) |
8 |
6 7
|
mpbi |
⊢ ( dom 𝐴 ∪ ( dom 𝐴 ∪ ran 𝐴 ) ) = ( dom 𝐴 ∪ ran 𝐴 ) |
9 |
3 5 8
|
3eqtri |
⊢ dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) = ( dom 𝐴 ∪ ran 𝐴 ) |
10 |
|
rnun |
⊢ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) = ( ran 𝐴 ∪ ran ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) |
11 |
|
rnresi |
⊢ ran ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) = ( dom 𝐴 ∪ ran 𝐴 ) |
12 |
11
|
uneq2i |
⊢ ( ran 𝐴 ∪ ran ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) = ( ran 𝐴 ∪ ( dom 𝐴 ∪ ran 𝐴 ) ) |
13 |
|
ssun2 |
⊢ ran 𝐴 ⊆ ( dom 𝐴 ∪ ran 𝐴 ) |
14 |
|
ssequn1 |
⊢ ( ran 𝐴 ⊆ ( dom 𝐴 ∪ ran 𝐴 ) ↔ ( ran 𝐴 ∪ ( dom 𝐴 ∪ ran 𝐴 ) ) = ( dom 𝐴 ∪ ran 𝐴 ) ) |
15 |
13 14
|
mpbi |
⊢ ( ran 𝐴 ∪ ( dom 𝐴 ∪ ran 𝐴 ) ) = ( dom 𝐴 ∪ ran 𝐴 ) |
16 |
10 12 15
|
3eqtri |
⊢ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) = ( dom 𝐴 ∪ ran 𝐴 ) |
17 |
9 16
|
uneq12i |
⊢ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) = ( ( dom 𝐴 ∪ ran 𝐴 ) ∪ ( dom 𝐴 ∪ ran 𝐴 ) ) |
18 |
|
unidm |
⊢ ( ( dom 𝐴 ∪ ran 𝐴 ) ∪ ( dom 𝐴 ∪ ran 𝐴 ) ) = ( dom 𝐴 ∪ ran 𝐴 ) |
19 |
17 18
|
eqtri |
⊢ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) = ( dom 𝐴 ∪ ran 𝐴 ) |
20 |
19
|
reseq2i |
⊢ ( I ↾ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) = ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) |
21 |
|
ssun2 |
⊢ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) |
22 |
20 21
|
eqsstri |
⊢ ( I ↾ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) |
23 |
1
|
elexi |
⊢ 𝐴 ∈ V |
24 |
|
dmexg |
⊢ ( 𝐴 ∈ 𝑉 → dom 𝐴 ∈ V ) |
25 |
|
rnexg |
⊢ ( 𝐴 ∈ 𝑉 → ran 𝐴 ∈ V ) |
26 |
|
unexg |
⊢ ( ( dom 𝐴 ∈ V ∧ ran 𝐴 ∈ V ) → ( dom 𝐴 ∪ ran 𝐴 ) ∈ V ) |
27 |
24 25 26
|
syl2anc |
⊢ ( 𝐴 ∈ 𝑉 → ( dom 𝐴 ∪ ran 𝐴 ) ∈ V ) |
28 |
27
|
resiexd |
⊢ ( 𝐴 ∈ 𝑉 → ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ∈ V ) |
29 |
1 28
|
ax-mp |
⊢ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ∈ V |
30 |
23 29
|
unex |
⊢ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∈ V |
31 |
|
dmeq |
⊢ ( 𝑥 = ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) → dom 𝑥 = dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) |
32 |
|
rneq |
⊢ ( 𝑥 = ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) → ran 𝑥 = ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) |
33 |
31 32
|
uneq12d |
⊢ ( 𝑥 = ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) → ( dom 𝑥 ∪ ran 𝑥 ) = ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) |
34 |
33
|
reseq2d |
⊢ ( 𝑥 = ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) → ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) = ( I ↾ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) ) |
35 |
|
id |
⊢ ( 𝑥 = ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) → 𝑥 = ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) |
36 |
34 35
|
sseq12d |
⊢ ( 𝑥 = ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) → ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑥 ↔ ( I ↾ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) |
37 |
36
|
cleq2lem |
⊢ ( 𝑥 = ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) → ( ( 𝐴 ⊆ 𝑥 ∧ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑥 ) ↔ ( 𝐴 ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∧ ( I ↾ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) ) |
38 |
30 37
|
spcev |
⊢ ( ( 𝐴 ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∧ ( I ↾ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) → ∃ 𝑥 ( 𝐴 ⊆ 𝑥 ∧ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑥 ) ) |
39 |
|
intexab |
⊢ ( ∃ 𝑥 ( 𝐴 ⊆ 𝑥 ∧ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑥 ) ↔ ∩ { 𝑥 ∣ ( 𝐴 ⊆ 𝑥 ∧ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑥 ) } ∈ V ) |
40 |
38 39
|
sylib |
⊢ ( ( 𝐴 ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∧ ( I ↾ ( dom ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ∪ ran ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) ) ⊆ ( 𝐴 ∪ ( I ↾ ( dom 𝐴 ∪ ran 𝐴 ) ) ) ) → ∩ { 𝑥 ∣ ( 𝐴 ⊆ 𝑥 ∧ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑥 ) } ∈ V ) |
41 |
2 22 40
|
mp2an |
⊢ ∩ { 𝑥 ∣ ( 𝐴 ⊆ 𝑥 ∧ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑥 ) } ∈ V |