Step | Hyp | Ref
| Expression |
1 | | catcxpccl.o |
. . . . 5
⊢ 𝑇 = (𝑋 ×_{c} 𝑌) |
2 | | eqid 2820 |
. . . . 5
⊢
(Base‘𝑋) =
(Base‘𝑋) |
3 | | eqid 2820 |
. . . . 5
⊢
(Base‘𝑌) =
(Base‘𝑌) |
4 | | eqid 2820 |
. . . . 5
⊢ (Hom
‘𝑋) = (Hom
‘𝑋) |
5 | | eqid 2820 |
. . . . 5
⊢ (Hom
‘𝑌) = (Hom
‘𝑌) |
6 | | eqid 2820 |
. . . . 5
⊢
(comp‘𝑋) =
(comp‘𝑋) |
7 | | eqid 2820 |
. . . . 5
⊢
(comp‘𝑌) =
(comp‘𝑌) |
8 | | catcxpccl.x |
. . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐵) |
9 | | catcxpccl.y |
. . . . 5
⊢ (𝜑 → 𝑌 ∈ 𝐵) |
10 | | eqidd 2821 |
. . . . 5
⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌))) |
11 | 1, 2, 3 | xpcbas 17407 |
. . . . . . 7
⊢
((Base‘𝑋)
× (Base‘𝑌)) =
(Base‘𝑇) |
12 | | eqid 2820 |
. . . . . . 7
⊢ (Hom
‘𝑇) = (Hom
‘𝑇) |
13 | 1, 11, 4, 5, 12 | xpchomfval 17408 |
. . . . . 6
⊢ (Hom
‘𝑇) = (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)))) |
14 | 13 | a1i 11 |
. . . . 5
⊢ (𝜑 → (Hom ‘𝑇) = (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))))) |
15 | | eqidd 2821 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩)) = (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩))) |
16 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
14, 15 | xpcval 17406 |
. . . 4
⊢ (𝜑 → 𝑇 = {⟨(Base‘ndx),
((Base‘𝑋) ×
(Base‘𝑌))⟩,
⟨(Hom ‘ndx), (Hom ‘𝑇)⟩, ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩))⟩}) |
17 | | catcxpccl.u |
. . . . 5
⊢ (𝜑 → 𝑈 ∈ WUni) |
18 | | df-base 16468 |
. . . . . . 7
⊢ Base =
Slot 1 |
19 | | catcxpccl.1 |
. . . . . . . 8
⊢ (𝜑 → ω ∈ 𝑈) |
20 | 17, 19 | wunndx 16483 |
. . . . . . 7
⊢ (𝜑 → ndx ∈ 𝑈) |
21 | 18, 17, 20 | wunstr 16486 |
. . . . . 6
⊢ (𝜑 → (Base‘ndx) ∈
𝑈) |
22 | | catcxpccl.c |
. . . . . . . . . . 11
⊢ 𝐶 = (CatCat‘𝑈) |
23 | | catcxpccl.b |
. . . . . . . . . . 11
⊢ 𝐵 = (Base‘𝐶) |
24 | 22, 23, 17 | catcbas 17336 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat)) |
25 | 8, 24 | eleqtrd 2913 |
. . . . . . . . 9
⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat)) |
26 | 25 | elin1d 4153 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ 𝑈) |
27 | 18, 17, 26 | wunstr 16486 |
. . . . . . 7
⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈) |
28 | 9, 24 | eleqtrd 2913 |
. . . . . . . . 9
⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat)) |
29 | 28 | elin1d 4153 |
. . . . . . . 8
⊢ (𝜑 → 𝑌 ∈ 𝑈) |
30 | 18, 17, 29 | wunstr 16486 |
. . . . . . 7
⊢ (𝜑 → (Base‘𝑌) ∈ 𝑈) |
31 | 17, 27, 30 | wunxp 10124 |
. . . . . 6
⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈) |
32 | 17, 21, 31 | wunop 10122 |
. . . . 5
⊢ (𝜑 → ⟨(Base‘ndx),
((Base‘𝑋) ×
(Base‘𝑌))⟩
∈ 𝑈) |
33 | | df-hom 16568 |
. . . . . . 7
⊢ Hom =
Slot ;14 |
34 | 33, 17, 20 | wunstr 16486 |
. . . . . 6
⊢ (𝜑 → (Hom ‘ndx) ∈
𝑈) |
35 | 17, 31, 31 | wunxp 10124 |
. . . . . . . 8
⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈) |
36 | 33, 17, 26 | wunstr 16486 |
. . . . . . . . . . . 12
⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈) |
37 | 17, 36 | wunrn 10129 |
. . . . . . . . . . 11
⊢ (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈) |
38 | 17, 37 | wununi 10106 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ ran (Hom ‘𝑋) ∈ 𝑈) |
39 | 33, 17, 29 | wunstr 16486 |
. . . . . . . . . . . 12
⊢ (𝜑 → (Hom ‘𝑌) ∈ 𝑈) |
40 | 17, 39 | wunrn 10129 |
. . . . . . . . . . 11
⊢ (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈) |
41 | 17, 40 | wununi 10106 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ ran (Hom ‘𝑌) ∈ 𝑈) |
42 | 17, 38, 41 | wunxp 10124 |
. . . . . . . . 9
⊢ (𝜑 → (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌)) ∈
𝑈) |
43 | 17, 42 | wunpw 10107 |
. . . . . . . 8
⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌)) ∈
𝑈) |
44 | | ovssunirn 7169 |
. . . . . . . . . . . . 13
⊢
((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) ⊆ ∪ ran
(Hom ‘𝑋) |
45 | | ovssunirn 7169 |
. . . . . . . . . . . . 13
⊢
((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)) ⊆ ∪ ran
(Hom ‘𝑌) |
46 | | xpss12 5546 |
. . . . . . . . . . . . 13
⊢
((((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) ⊆ ∪ ran
(Hom ‘𝑋) ∧
((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)) ⊆ ∪ ran
(Hom ‘𝑌)) →
(((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))) ⊆ (∪ ran
(Hom ‘𝑋) ×
∪ ran (Hom ‘𝑌))) |
47 | 44, 45, 46 | mp2an 690 |
. . . . . . . . . . . 12
⊢
(((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))) ⊆ (∪ ran
(Hom ‘𝑋) ×
∪ ran (Hom ‘𝑌)) |
48 | | ovex 7166 |
. . . . . . . . . . . . . 14
⊢
((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) ∈ V |
49 | | ovex 7166 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)) ∈ V |
50 | 48, 49 | xpex 7454 |
. . . . . . . . . . . . 13
⊢
(((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))) ∈ V |
51 | 50 | elpw 4519 |
. . . . . . . . . . . 12
⊢
((((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌)) ↔
(((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))) ⊆ (∪ ran
(Hom ‘𝑋) ×
∪ ran (Hom ‘𝑌))) |
52 | 47, 51 | mpbir 233 |
. . . . . . . . . . 11
⊢
(((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌)) |
53 | 52 | rgen2w 3138 |
. . . . . . . . . 10
⊢
∀𝑢 ∈
((Base‘𝑋) ×
(Base‘𝑌))∀𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌))(((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌)) |
54 | | eqid 2820 |
. . . . . . . . . . 11
⊢ (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)))) = (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)))) |
55 | 54 | fmpo 7744 |
. . . . . . . . . 10
⊢
(∀𝑢 ∈
((Base‘𝑋) ×
(Base‘𝑌))∀𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌))(((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌)) ↔
(𝑢 ∈
((Base‘𝑋) ×
(Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1^{st}
‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd}
‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)))):(((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌)))⟶𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌))) |
56 | 53, 55 | mpbi 232 |
. . . . . . . . 9
⊢ (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)))):(((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌)))⟶𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌)) |
57 | 56 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)))):(((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌)))⟶𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran
(Hom ‘𝑌))) |
58 | 17, 35, 43, 57 | wunf 10127 |
. . . . . . 7
⊢ (𝜑 → (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1^{st} ‘𝑢)(Hom ‘𝑋)(1^{st} ‘𝑣)) × ((2^{nd} ‘𝑢)(Hom ‘𝑌)(2^{nd} ‘𝑣)))) ∈ 𝑈) |
59 | 13, 58 | eqeltrid 2915 |
. . . . . 6
⊢ (𝜑 → (Hom ‘𝑇) ∈ 𝑈) |
60 | 17, 34, 59 | wunop 10122 |
. . . . 5
⊢ (𝜑 → ⟨(Hom ‘ndx),
(Hom ‘𝑇)⟩ ∈
𝑈) |
61 | | df-cco 16569 |
. . . . . . 7
⊢ comp =
Slot ;15 |
62 | 61, 17, 20 | wunstr 16486 |
. . . . . 6
⊢ (𝜑 → (comp‘ndx) ∈
𝑈) |
63 | 17, 35, 31 | wunxp 10124 |
. . . . . . 7
⊢ (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈) |
64 | 61, 17, 26 | wunstr 16486 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈) |
65 | 17, 64 | wunrn 10129 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈) |
66 | 17, 65 | wununi 10106 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈) |
67 | 17, 66 | wunrn 10129 |
. . . . . . . . . . 11
⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈) |
68 | 17, 67 | wununi 10106 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈) |
69 | 17, 68 | wunpw 10107 |
. . . . . . . . 9
⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈) |
70 | 61, 17, 29 | wunstr 16486 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈) |
71 | 17, 70 | wunrn 10129 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈) |
72 | 17, 71 | wununi 10106 |
. . . . . . . . . . . 12
⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈) |
73 | 17, 72 | wunrn 10129 |
. . . . . . . . . . 11
⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈) |
74 | 17, 73 | wununi 10106 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈) |
75 | 17, 74 | wunpw 10107 |
. . . . . . . . 9
⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈) |
76 | 17, 69, 75 | wunxp 10124 |
. . . . . . . 8
⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ 𝑈) |
77 | 17, 59 | wunrn 10129 |
. . . . . . . . . 10
⊢ (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈) |
78 | 17, 77 | wununi 10106 |
. . . . . . . . 9
⊢ (𝜑 → ∪ ran (Hom ‘𝑇) ∈ 𝑈) |
79 | 17, 78, 78 | wunxp 10124 |
. . . . . . . 8
⊢ (𝜑 → (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇)) ∈
𝑈) |
80 | 17, 76, 79 | wunpm 10125 |
. . . . . . 7
⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑_{pm} (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇))) ∈
𝑈) |
81 | | fvex 6659 |
. . . . . . . . . . . . . . . . 17
⊢
(comp‘𝑋)
∈ V |
82 | 81 | rnex 7595 |
. . . . . . . . . . . . . . . 16
⊢ ran
(comp‘𝑋) ∈
V |
83 | 82 | uniex 7445 |
. . . . . . . . . . . . . . 15
⊢ ∪ ran (comp‘𝑋) ∈ V |
84 | 83 | rnex 7595 |
. . . . . . . . . . . . . 14
⊢ ran ∪ ran (comp‘𝑋) ∈ V |
85 | 84 | uniex 7445 |
. . . . . . . . . . . . 13
⊢ ∪ ran ∪ ran (comp‘𝑋) ∈ V |
86 | 85 | pwex 5257 |
. . . . . . . . . . . 12
⊢ 𝒫
∪ ran ∪ ran
(comp‘𝑋) ∈
V |
87 | | fvex 6659 |
. . . . . . . . . . . . . . . . 17
⊢
(comp‘𝑌)
∈ V |
88 | 87 | rnex 7595 |
. . . . . . . . . . . . . . . 16
⊢ ran
(comp‘𝑌) ∈
V |
89 | 88 | uniex 7445 |
. . . . . . . . . . . . . . 15
⊢ ∪ ran (comp‘𝑌) ∈ V |
90 | 89 | rnex 7595 |
. . . . . . . . . . . . . 14
⊢ ran ∪ ran (comp‘𝑌) ∈ V |
91 | 90 | uniex 7445 |
. . . . . . . . . . . . 13
⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V |
92 | 91 | pwex 5257 |
. . . . . . . . . . . 12
⊢ 𝒫
∪ ran ∪ ran
(comp‘𝑌) ∈
V |
93 | 86, 92 | xpex 7454 |
. . . . . . . . . . 11
⊢
(𝒫 ∪ ran ∪
ran (comp‘𝑋) ×
𝒫 ∪ ran ∪ ran
(comp‘𝑌)) ∈
V |
94 | | fvex 6659 |
. . . . . . . . . . . . . 14
⊢ (Hom
‘𝑇) ∈
V |
95 | 94 | rnex 7595 |
. . . . . . . . . . . . 13
⊢ ran (Hom
‘𝑇) ∈
V |
96 | 95 | uniex 7445 |
. . . . . . . . . . . 12
⊢ ∪ ran (Hom ‘𝑇) ∈ V |
97 | 96, 96 | xpex 7454 |
. . . . . . . . . . 11
⊢ (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇)) ∈
V |
98 | | ovssunirn 7169 |
. . . . . . . . . . . . . . . 16
⊢
((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)) ⊆ ∪ ran
(⟨(1^{st} ‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦)) |
99 | | ovssunirn 7169 |
. . . . . . . . . . . . . . . . 17
⊢
(⟨(1^{st} ‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦)) ⊆ ∪ ran
(comp‘𝑋) |
100 | | rnss 5785 |
. . . . . . . . . . . . . . . . 17
⊢
((⟨(1^{st} ‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦)) ⊆ ∪ ran
(comp‘𝑋) → ran
(⟨(1^{st} ‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦)) ⊆ ran ∪
ran (comp‘𝑋)) |
101 | | uniss 4822 |
. . . . . . . . . . . . . . . . 17
⊢ (ran
(⟨(1^{st} ‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦)) ⊆ ran ∪
ran (comp‘𝑋) →
∪ ran (⟨(1^{st} ‘(1^{st}
‘𝑥)), (1^{st}
‘(2^{nd} ‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦)) ⊆ ∪ ran
∪ ran (comp‘𝑋)) |
102 | 99, 100, 101 | mp2b 10 |
. . . . . . . . . . . . . . . 16
⊢ ∪ ran (⟨(1^{st} ‘(1^{st}
‘𝑥)), (1^{st}
‘(2^{nd} ‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦)) ⊆ ∪ ran
∪ ran (comp‘𝑋) |
103 | 98, 102 | sstri 3955 |
. . . . . . . . . . . . . . 15
⊢
((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)) ⊆ ∪ ran
∪ ran (comp‘𝑋) |
104 | | ovex 7166 |
. . . . . . . . . . . . . . . 16
⊢
((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)) ∈ V |
105 | 104 | elpw 4519 |
. . . . . . . . . . . . . . 15
⊢
(((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑋) ↔ ((1^{st}
‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)) ⊆ ∪ ran
∪ ran (comp‘𝑋)) |
106 | 103, 105 | mpbir 233 |
. . . . . . . . . . . . . 14
⊢
((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑋) |
107 | | ovssunirn 7169 |
. . . . . . . . . . . . . . . 16
⊢
((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓)) ⊆ ∪ ran
(⟨(2^{nd} ‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦)) |
108 | | ovssunirn 7169 |
. . . . . . . . . . . . . . . . 17
⊢
(⟨(2^{nd} ‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦)) ⊆ ∪ ran
(comp‘𝑌) |
109 | | rnss 5785 |
. . . . . . . . . . . . . . . . 17
⊢
((⟨(2^{nd} ‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦)) ⊆ ∪ ran
(comp‘𝑌) → ran
(⟨(2^{nd} ‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦)) ⊆ ran ∪
ran (comp‘𝑌)) |
110 | | uniss 4822 |
. . . . . . . . . . . . . . . . 17
⊢ (ran
(⟨(2^{nd} ‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦)) ⊆ ran ∪
ran (comp‘𝑌) →
∪ ran (⟨(2^{nd} ‘(1^{st}
‘𝑥)), (2^{nd}
‘(2^{nd} ‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦)) ⊆ ∪ ran
∪ ran (comp‘𝑌)) |
111 | 108, 109,
110 | mp2b 10 |
. . . . . . . . . . . . . . . 16
⊢ ∪ ran (⟨(2^{nd} ‘(1^{st}
‘𝑥)), (2^{nd}
‘(2^{nd} ‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦)) ⊆ ∪ ran
∪ ran (comp‘𝑌) |
112 | 107, 111 | sstri 3955 |
. . . . . . . . . . . . . . 15
⊢
((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓)) ⊆ ∪ ran
∪ ran (comp‘𝑌) |
113 | | ovex 7166 |
. . . . . . . . . . . . . . . 16
⊢
((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓)) ∈ V |
114 | 113 | elpw 4519 |
. . . . . . . . . . . . . . 15
⊢
(((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ↔ ((2^{nd}
‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓)) ⊆ ∪ ran
∪ ran (comp‘𝑌)) |
115 | 112, 114 | mpbir 233 |
. . . . . . . . . . . . . 14
⊢
((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌) |
116 | | opelxpi 5568 |
. . . . . . . . . . . . . 14
⊢
((((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∧ ((2^{nd}
‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌)) → ⟨((1^{st}
‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩ ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌))) |
117 | 106, 115,
116 | mp2an 690 |
. . . . . . . . . . . . 13
⊢
⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩ ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) |
118 | 117 | rgen2w 3138 |
. . . . . . . . . . . 12
⊢
∀𝑔 ∈
((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦)∀𝑓 ∈ ((Hom ‘𝑇)‘𝑥)⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩ ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) |
119 | | eqid 2820 |
. . . . . . . . . . . . 13
⊢ (𝑔 ∈ ((2^{nd}
‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩) = (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩) |
120 | 119 | fmpo 7744 |
. . . . . . . . . . . 12
⊢
(∀𝑔 ∈
((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦)∀𝑓 ∈ ((Hom ‘𝑇)‘𝑥)⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩ ∈ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↔ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩):(((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌))) |
121 | 118, 120 | mpbi 232 |
. . . . . . . . . . 11
⊢ (𝑔 ∈ ((2^{nd}
‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩):(((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) |
122 | | ovssunirn 7169 |
. . . . . . . . . . . 12
⊢
((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦) ⊆ ∪ ran
(Hom ‘𝑇) |
123 | | fvssunirn 6675 |
. . . . . . . . . . . 12
⊢ ((Hom
‘𝑇)‘𝑥) ⊆ ∪ ran (Hom ‘𝑇) |
124 | | xpss12 5546 |
. . . . . . . . . . . 12
⊢
((((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦) ⊆ ∪ ran
(Hom ‘𝑇) ∧ ((Hom
‘𝑇)‘𝑥) ⊆ ∪ ran (Hom ‘𝑇)) → (((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ (∪ ran
(Hom ‘𝑇) ×
∪ ran (Hom ‘𝑇))) |
125 | 122, 123,
124 | mp2an 690 |
. . . . . . . . . . 11
⊢
(((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ (∪ ran
(Hom ‘𝑇) ×
∪ ran (Hom ‘𝑇)) |
126 | | elpm2r 8402 |
. . . . . . . . . . 11
⊢
((((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ V ∧ (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇)) ∈ V)
∧ ((𝑔 ∈
((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩):(((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥))⟶(𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∧ (((2^{nd}
‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ (∪ ran
(Hom ‘𝑇) ×
∪ ran (Hom ‘𝑇)))) → (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑_{pm} (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇)))) |
127 | 93, 97, 121, 125, 126 | mp4an 691 |
. . . . . . . . . 10
⊢ (𝑔 ∈ ((2^{nd}
‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑_{pm} (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇))) |
128 | 127 | rgen2w 3138 |
. . . . . . . . 9
⊢
∀𝑥 ∈
(((Base‘𝑋) ×
(Base‘𝑌)) ×
((Base‘𝑋) ×
(Base‘𝑌)))∀𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌))(𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑_{pm} (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇))) |
129 | | eqid 2820 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩)) = (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩)) |
130 | 129 | fmpo 7744 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
(((Base‘𝑋) ×
(Base‘𝑌)) ×
((Base‘𝑋) ×
(Base‘𝑌)))∀𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌))(𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩) ∈ ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑_{pm} (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇))) ↔
(𝑥 ∈
(((Base‘𝑋) ×
(Base‘𝑌)) ×
((Base‘𝑋) ×
(Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩)):((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌)))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑_{pm} (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇)))) |
131 | 128, 130 | mpbi 232 |
. . . . . . . 8
⊢ (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩)):((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌)))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑_{pm} (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇))) |
132 | 131 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩)):((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌)))⟶((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑_{pm} (∪ ran (Hom ‘𝑇) × ∪ ran
(Hom ‘𝑇)))) |
133 | 17, 63, 80, 132 | wunf 10127 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩)) ∈ 𝑈) |
134 | 17, 62, 133 | wunop 10122 |
. . . . 5
⊢ (𝜑 → ⟨(comp‘ndx),
(𝑥 ∈
(((Base‘𝑋) ×
(Base‘𝑌)) ×
((Base‘𝑋) ×
(Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩))⟩ ∈ 𝑈) |
135 | 17, 32, 60, 134 | wuntp 10111 |
. . . 4
⊢ (𝜑 → {⟨(Base‘ndx),
((Base‘𝑋) ×
(Base‘𝑌))⟩,
⟨(Hom ‘ndx), (Hom ‘𝑇)⟩, ⟨(comp‘ndx), (𝑥 ∈ (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))), 𝑦 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (𝑔 ∈ ((2^{nd} ‘𝑥)(Hom ‘𝑇)𝑦), 𝑓 ∈ ((Hom ‘𝑇)‘𝑥) ↦ ⟨((1^{st} ‘𝑔)(⟨(1^{st}
‘(1^{st} ‘𝑥)), (1^{st} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑋)(1^{st} ‘𝑦))(1^{st} ‘𝑓)), ((2^{nd} ‘𝑔)(⟨(2^{nd}
‘(1^{st} ‘𝑥)), (2^{nd} ‘(2^{nd}
‘𝑥))⟩(comp‘𝑌)(2^{nd} ‘𝑦))(2^{nd} ‘𝑓))⟩))⟩} ∈ 𝑈) |
136 | 16, 135 | eqeltrd 2911 |
. . 3
⊢ (𝜑 → 𝑇 ∈ 𝑈) |
137 | 25 | elin2d 4154 |
. . . 4
⊢ (𝜑 → 𝑋 ∈ Cat) |
138 | 28 | elin2d 4154 |
. . . 4
⊢ (𝜑 → 𝑌 ∈ Cat) |
139 | 1, 137, 138 | xpccat 17419 |
. . 3
⊢ (𝜑 → 𝑇 ∈ Cat) |
140 | 136, 139 | elind 4149 |
. 2
⊢ (𝜑 → 𝑇 ∈ (𝑈 ∩ Cat)) |
141 | 140, 24 | eleqtrrd 2914 |
1
⊢ (𝜑 → 𝑇 ∈ 𝐵) |