| Step |
Hyp |
Ref |
Expression |
| 1 |
|
df-rtrcl |
⊢ t* = ( 𝑥 ∈ V ↦ ∩ { 𝑧 ∣ ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 2 |
|
ancom |
⊢ ( ( ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ∧ ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ) ↔ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) |
| 3 |
2
|
anbi2i |
⊢ ( ( 𝑥 ⊆ 𝑦 ∧ ( ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ∧ ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ) ) ↔ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) ) |
| 4 |
3
|
abbii |
⊢ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ∧ ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ) ) } = { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } |
| 5 |
4
|
inteqi |
⊢ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ∧ ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ) ) } = ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } |
| 6 |
5
|
mpteq2i |
⊢ ( 𝑥 ∈ V ↦ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ∧ ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ) ) } ) = ( 𝑥 ∈ V ↦ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) |
| 7 |
|
vex |
⊢ 𝑥 ∈ V |
| 8 |
7
|
rtrclexi |
⊢ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ∈ V |
| 9 |
8
|
a1i |
⊢ ( 𝑥 ∈ V → ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ∈ V ) |
| 10 |
|
dmexg |
⊢ ( 𝑥 ∈ V → dom 𝑥 ∈ V ) |
| 11 |
|
rnexg |
⊢ ( 𝑥 ∈ V → ran 𝑥 ∈ V ) |
| 12 |
10 11
|
unexd |
⊢ ( 𝑥 ∈ V → ( dom 𝑥 ∪ ran 𝑥 ) ∈ V ) |
| 13 |
|
resiexg |
⊢ ( ( dom 𝑥 ∪ ran 𝑥 ) ∈ V → ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ∈ V ) |
| 14 |
7 12 13
|
mp2b |
⊢ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ∈ V |
| 15 |
7 14
|
unex |
⊢ ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ∈ V |
| 16 |
15
|
trclexi |
⊢ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∈ V |
| 17 |
16
|
a1i |
⊢ ( 𝑥 ∈ V → ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∈ V ) |
| 18 |
|
simpr |
⊢ ( ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) → ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) |
| 19 |
18
|
cotrintab |
⊢ ( ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∘ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ⊆ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } |
| 20 |
19
|
a1i |
⊢ ( 𝑥 ∈ V → ( ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∘ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ⊆ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 21 |
7
|
dmex |
⊢ dom 𝑥 ∈ V |
| 22 |
7
|
rnex |
⊢ ran 𝑥 ∈ V |
| 23 |
|
unexg |
⊢ ( ( dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V ) → ( dom 𝑥 ∪ ran 𝑥 ) ∈ V ) |
| 24 |
23
|
resiexd |
⊢ ( ( dom 𝑥 ∈ V ∧ ran 𝑥 ∈ V ) → ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ∈ V ) |
| 25 |
21 22 24
|
mp2an |
⊢ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ∈ V |
| 26 |
7 25
|
unex |
⊢ ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ∈ V |
| 27 |
|
dmtrcl |
⊢ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ∈ V → dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } = dom ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ) |
| 28 |
26 27
|
ax-mp |
⊢ dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } = dom ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) |
| 29 |
|
dmun |
⊢ dom ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) = ( dom 𝑥 ∪ dom ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) |
| 30 |
|
dmresi |
⊢ dom ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) = ( dom 𝑥 ∪ ran 𝑥 ) |
| 31 |
30
|
uneq2i |
⊢ ( dom 𝑥 ∪ dom ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) = ( dom 𝑥 ∪ ( dom 𝑥 ∪ ran 𝑥 ) ) |
| 32 |
|
ssun1 |
⊢ dom 𝑥 ⊆ ( dom 𝑥 ∪ ran 𝑥 ) |
| 33 |
|
ssequn1 |
⊢ ( dom 𝑥 ⊆ ( dom 𝑥 ∪ ran 𝑥 ) ↔ ( dom 𝑥 ∪ ( dom 𝑥 ∪ ran 𝑥 ) ) = ( dom 𝑥 ∪ ran 𝑥 ) ) |
| 34 |
32 33
|
mpbi |
⊢ ( dom 𝑥 ∪ ( dom 𝑥 ∪ ran 𝑥 ) ) = ( dom 𝑥 ∪ ran 𝑥 ) |
| 35 |
29 31 34
|
3eqtri |
⊢ dom ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) = ( dom 𝑥 ∪ ran 𝑥 ) |
| 36 |
28 35
|
eqtri |
⊢ dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } = ( dom 𝑥 ∪ ran 𝑥 ) |
| 37 |
|
rntrcl |
⊢ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ∈ V → ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } = ran ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ) |
| 38 |
26 37
|
ax-mp |
⊢ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } = ran ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) |
| 39 |
|
rnun |
⊢ ran ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) = ( ran 𝑥 ∪ ran ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) |
| 40 |
|
rnresi |
⊢ ran ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) = ( dom 𝑥 ∪ ran 𝑥 ) |
| 41 |
40
|
uneq2i |
⊢ ( ran 𝑥 ∪ ran ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) = ( ran 𝑥 ∪ ( dom 𝑥 ∪ ran 𝑥 ) ) |
| 42 |
|
ssun2 |
⊢ ran 𝑥 ⊆ ( dom 𝑥 ∪ ran 𝑥 ) |
| 43 |
|
ssequn1 |
⊢ ( ran 𝑥 ⊆ ( dom 𝑥 ∪ ran 𝑥 ) ↔ ( ran 𝑥 ∪ ( dom 𝑥 ∪ ran 𝑥 ) ) = ( dom 𝑥 ∪ ran 𝑥 ) ) |
| 44 |
42 43
|
mpbi |
⊢ ( ran 𝑥 ∪ ( dom 𝑥 ∪ ran 𝑥 ) ) = ( dom 𝑥 ∪ ran 𝑥 ) |
| 45 |
39 41 44
|
3eqtri |
⊢ ran ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) = ( dom 𝑥 ∪ ran 𝑥 ) |
| 46 |
38 45
|
eqtri |
⊢ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } = ( dom 𝑥 ∪ ran 𝑥 ) |
| 47 |
36 46
|
uneq12i |
⊢ ( dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∪ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) = ( ( dom 𝑥 ∪ ran 𝑥 ) ∪ ( dom 𝑥 ∪ ran 𝑥 ) ) |
| 48 |
|
unidm |
⊢ ( ( dom 𝑥 ∪ ran 𝑥 ) ∪ ( dom 𝑥 ∪ ran 𝑥 ) ) = ( dom 𝑥 ∪ ran 𝑥 ) |
| 49 |
47 48
|
eqtri |
⊢ ( dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∪ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) = ( dom 𝑥 ∪ ran 𝑥 ) |
| 50 |
49
|
reseq2i |
⊢ ( I ↾ ( dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∪ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) = ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) |
| 51 |
|
ssun2 |
⊢ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) |
| 52 |
|
ssmin |
⊢ ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } |
| 53 |
51 52
|
sstri |
⊢ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } |
| 54 |
50 53
|
eqsstri |
⊢ ( I ↾ ( dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∪ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) ⊆ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } |
| 55 |
54
|
a1i |
⊢ ( 𝑥 ∈ V → ( I ↾ ( dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∪ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) ⊆ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 56 |
|
simprl |
⊢ ( ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) → ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ) |
| 57 |
56
|
cotrintab |
⊢ ( ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ∘ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) ⊆ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } |
| 58 |
57
|
a1i |
⊢ ( 𝑥 ∈ V → ( ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ∘ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) ⊆ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) |
| 59 |
|
id |
⊢ ( 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } → 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 60 |
59 59
|
coeq12d |
⊢ ( 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } → ( 𝑦 ∘ 𝑦 ) = ( ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∘ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) |
| 61 |
60 59
|
sseq12d |
⊢ ( 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } → ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ↔ ( ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∘ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ⊆ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) |
| 62 |
|
dmeq |
⊢ ( 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } → dom 𝑦 = dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 63 |
|
rneq |
⊢ ( 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } → ran 𝑦 = ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 64 |
62 63
|
uneq12d |
⊢ ( 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } → ( dom 𝑦 ∪ ran 𝑦 ) = ( dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∪ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) |
| 65 |
64
|
reseq2d |
⊢ ( 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } → ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) = ( I ↾ ( dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∪ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) ) |
| 66 |
65 59
|
sseq12d |
⊢ ( 𝑦 = ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } → ( ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ↔ ( I ↾ ( dom ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ∪ ran ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) ⊆ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) ) |
| 67 |
|
id |
⊢ ( 𝑧 = ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } → 𝑧 = ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) |
| 68 |
67 67
|
coeq12d |
⊢ ( 𝑧 = ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } → ( 𝑧 ∘ 𝑧 ) = ( ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ∘ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) ) |
| 69 |
68 67
|
sseq12d |
⊢ ( 𝑧 = ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } → ( ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ↔ ( ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ∘ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) ⊆ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) ) |
| 70 |
9 17 20 55 58 61 66 69
|
mptrcllem |
⊢ ( 𝑥 ∈ V ↦ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ∧ ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ) ) } ) = ( 𝑥 ∈ V ↦ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 71 |
|
df-3an |
⊢ ( ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) ↔ ( ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ) ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) ) |
| 72 |
|
ancom |
⊢ ( ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ) ↔ ( 𝑥 ⊆ 𝑧 ∧ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ) ) |
| 73 |
|
unss |
⊢ ( ( 𝑥 ⊆ 𝑧 ∧ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ) ↔ ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ) |
| 74 |
72 73
|
bitri |
⊢ ( ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ) ↔ ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ) |
| 75 |
74
|
anbi1i |
⊢ ( ( ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ) ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) ↔ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) ) |
| 76 |
71 75
|
bitr2i |
⊢ ( ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) ↔ ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) ) |
| 77 |
76
|
abbii |
⊢ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } = { 𝑧 ∣ ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } |
| 78 |
77
|
inteqi |
⊢ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } = ∩ { 𝑧 ∣ ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } |
| 79 |
78
|
mpteq2i |
⊢ ( 𝑥 ∈ V ↦ ∩ { 𝑧 ∣ ( ( 𝑥 ∪ ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ) ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) = ( 𝑥 ∈ V ↦ ∩ { 𝑧 ∣ ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 80 |
6 70 79
|
3eqtri |
⊢ ( 𝑥 ∈ V ↦ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ∧ ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ) ) } ) = ( 𝑥 ∈ V ↦ ∩ { 𝑧 ∣ ( ( I ↾ ( dom 𝑥 ∪ ran 𝑥 ) ) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ ( 𝑧 ∘ 𝑧 ) ⊆ 𝑧 ) } ) |
| 81 |
1 80
|
eqtr4i |
⊢ t* = ( 𝑥 ∈ V ↦ ∩ { 𝑦 ∣ ( 𝑥 ⊆ 𝑦 ∧ ( ( I ↾ ( dom 𝑦 ∪ ran 𝑦 ) ) ⊆ 𝑦 ∧ ( 𝑦 ∘ 𝑦 ) ⊆ 𝑦 ) ) } ) |