Step |
Hyp |
Ref |
Expression |
1 |
|
catcxpccl.c |
⊢ 𝐶 = ( CatCat ‘ 𝑈 ) |
2 |
|
catcxpccl.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
3 |
|
catcxpccl.o |
⊢ 𝑇 = ( 𝑋 ×c 𝑌 ) |
4 |
|
catcxpccl.u |
⊢ ( 𝜑 → 𝑈 ∈ WUni ) |
5 |
|
catcxpccl.1 |
⊢ ( 𝜑 → ω ∈ 𝑈 ) |
6 |
|
catcxpccl.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐵 ) |
7 |
|
catcxpccl.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝐵 ) |
8 |
|
eqid |
⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 ) |
9 |
|
eqid |
⊢ ( Base ‘ 𝑌 ) = ( Base ‘ 𝑌 ) |
10 |
|
eqid |
⊢ ( Hom ‘ 𝑋 ) = ( Hom ‘ 𝑋 ) |
11 |
|
eqid |
⊢ ( Hom ‘ 𝑌 ) = ( Hom ‘ 𝑌 ) |
12 |
|
eqid |
⊢ ( comp ‘ 𝑋 ) = ( comp ‘ 𝑋 ) |
13 |
|
eqid |
⊢ ( comp ‘ 𝑌 ) = ( comp ‘ 𝑌 ) |
14 |
|
eqidd |
⊢ ( 𝜑 → ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) = ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) |
15 |
3 8 9
|
xpcbas |
⊢ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) = ( Base ‘ 𝑇 ) |
16 |
|
eqid |
⊢ ( Hom ‘ 𝑇 ) = ( Hom ‘ 𝑇 ) |
17 |
3 15 10 11 16
|
xpchomfval |
⊢ ( Hom ‘ 𝑇 ) = ( 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ) |
18 |
17
|
a1i |
⊢ ( 𝜑 → ( Hom ‘ 𝑇 ) = ( 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ) ) |
19 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) , 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) = ( 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) , 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) ) |
20 |
3 8 9 10 11 12 13 6 7 14 18 19
|
xpcval |
⊢ ( 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 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 } ) |
21 |
|
baseid |
⊢ Base = Slot ( Base ‘ ndx ) |
22 |
4 5
|
wunndx |
⊢ ( 𝜑 → ndx ∈ 𝑈 ) |
23 |
21 4 22
|
wunstr |
⊢ ( 𝜑 → ( Base ‘ ndx ) ∈ 𝑈 ) |
24 |
1 2 4 6
|
catcbaselcl |
⊢ ( 𝜑 → ( Base ‘ 𝑋 ) ∈ 𝑈 ) |
25 |
1 2 4 7
|
catcbaselcl |
⊢ ( 𝜑 → ( Base ‘ 𝑌 ) ∈ 𝑈 ) |
26 |
4 24 25
|
wunxp |
⊢ ( 𝜑 → ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ∈ 𝑈 ) |
27 |
4 23 26
|
wunop |
⊢ ( 𝜑 → 〈 ( Base ‘ ndx ) , ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) 〉 ∈ 𝑈 ) |
28 |
|
homid |
⊢ Hom = Slot ( Hom ‘ ndx ) |
29 |
28 4 22
|
wunstr |
⊢ ( 𝜑 → ( Hom ‘ ndx ) ∈ 𝑈 ) |
30 |
4 26 26
|
wunxp |
⊢ ( 𝜑 → ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) ∈ 𝑈 ) |
31 |
1 2 4 6
|
catchomcl |
⊢ ( 𝜑 → ( Hom ‘ 𝑋 ) ∈ 𝑈 ) |
32 |
4 31
|
wunrn |
⊢ ( 𝜑 → ran ( Hom ‘ 𝑋 ) ∈ 𝑈 ) |
33 |
4 32
|
wununi |
⊢ ( 𝜑 → ∪ ran ( Hom ‘ 𝑋 ) ∈ 𝑈 ) |
34 |
1 2 4 7
|
catchomcl |
⊢ ( 𝜑 → ( Hom ‘ 𝑌 ) ∈ 𝑈 ) |
35 |
4 34
|
wunrn |
⊢ ( 𝜑 → ran ( Hom ‘ 𝑌 ) ∈ 𝑈 ) |
36 |
4 35
|
wununi |
⊢ ( 𝜑 → ∪ ran ( Hom ‘ 𝑌 ) ∈ 𝑈 ) |
37 |
4 33 36
|
wunxp |
⊢ ( 𝜑 → ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) ∈ 𝑈 ) |
38 |
4 37
|
wunpw |
⊢ ( 𝜑 → 𝒫 ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) ∈ 𝑈 ) |
39 |
|
ovssunirn |
⊢ ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) ⊆ ∪ ran ( Hom ‘ 𝑋 ) |
40 |
|
ovssunirn |
⊢ ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ⊆ ∪ ran ( Hom ‘ 𝑌 ) |
41 |
|
xpss12 |
⊢ ( ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) ⊆ ∪ ran ( Hom ‘ 𝑋 ) ∧ ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ⊆ ∪ ran ( Hom ‘ 𝑌 ) ) → ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ⊆ ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) ) |
42 |
39 40 41
|
mp2an |
⊢ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ⊆ ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) |
43 |
|
ovex |
⊢ ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) ∈ V |
44 |
|
ovex |
⊢ ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ∈ V |
45 |
43 44
|
xpex |
⊢ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ∈ V |
46 |
45
|
elpw |
⊢ ( ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ∈ 𝒫 ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) ↔ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ⊆ ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) ) |
47 |
42 46
|
mpbir |
⊢ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ∈ 𝒫 ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) |
48 |
47
|
rgen2w |
⊢ ∀ 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ∀ 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ∈ 𝒫 ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) |
49 |
|
eqid |
⊢ ( 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ) = ( 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ) |
50 |
49
|
fmpo |
⊢ ( ∀ 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ∀ 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ∈ 𝒫 ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) ↔ ( 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ) : ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) ⟶ 𝒫 ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) ) |
51 |
48 50
|
mpbi |
⊢ ( 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ) : ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) ⟶ 𝒫 ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) |
52 |
51
|
a1i |
⊢ ( 𝜑 → ( 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ) : ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) ⟶ 𝒫 ( ∪ ran ( Hom ‘ 𝑋 ) × ∪ ran ( Hom ‘ 𝑌 ) ) ) |
53 |
4 30 38 52
|
wunf |
⊢ ( 𝜑 → ( 𝑢 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) , 𝑣 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( ( ( 1st ‘ 𝑢 ) ( Hom ‘ 𝑋 ) ( 1st ‘ 𝑣 ) ) × ( ( 2nd ‘ 𝑢 ) ( Hom ‘ 𝑌 ) ( 2nd ‘ 𝑣 ) ) ) ) ∈ 𝑈 ) |
54 |
17 53
|
eqeltrid |
⊢ ( 𝜑 → ( Hom ‘ 𝑇 ) ∈ 𝑈 ) |
55 |
4 29 54
|
wunop |
⊢ ( 𝜑 → 〈 ( Hom ‘ ndx ) , ( Hom ‘ 𝑇 ) 〉 ∈ 𝑈 ) |
56 |
|
ccoid |
⊢ comp = Slot ( comp ‘ ndx ) |
57 |
56 4 22
|
wunstr |
⊢ ( 𝜑 → ( comp ‘ ndx ) ∈ 𝑈 ) |
58 |
4 30 26
|
wunxp |
⊢ ( 𝜑 → ( ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) ∈ 𝑈 ) |
59 |
1 2 4 6
|
catcccocl |
⊢ ( 𝜑 → ( comp ‘ 𝑋 ) ∈ 𝑈 ) |
60 |
4 59
|
wunrn |
⊢ ( 𝜑 → ran ( comp ‘ 𝑋 ) ∈ 𝑈 ) |
61 |
4 60
|
wununi |
⊢ ( 𝜑 → ∪ ran ( comp ‘ 𝑋 ) ∈ 𝑈 ) |
62 |
4 61
|
wunrn |
⊢ ( 𝜑 → ran ∪ ran ( comp ‘ 𝑋 ) ∈ 𝑈 ) |
63 |
4 62
|
wununi |
⊢ ( 𝜑 → ∪ ran ∪ ran ( comp ‘ 𝑋 ) ∈ 𝑈 ) |
64 |
4 63
|
wunpw |
⊢ ( 𝜑 → 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) ∈ 𝑈 ) |
65 |
1 2 4 7
|
catcccocl |
⊢ ( 𝜑 → ( comp ‘ 𝑌 ) ∈ 𝑈 ) |
66 |
4 65
|
wunrn |
⊢ ( 𝜑 → ran ( comp ‘ 𝑌 ) ∈ 𝑈 ) |
67 |
4 66
|
wununi |
⊢ ( 𝜑 → ∪ ran ( comp ‘ 𝑌 ) ∈ 𝑈 ) |
68 |
4 67
|
wunrn |
⊢ ( 𝜑 → ran ∪ ran ( comp ‘ 𝑌 ) ∈ 𝑈 ) |
69 |
4 68
|
wununi |
⊢ ( 𝜑 → ∪ ran ∪ ran ( comp ‘ 𝑌 ) ∈ 𝑈 ) |
70 |
4 69
|
wunpw |
⊢ ( 𝜑 → 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ∈ 𝑈 ) |
71 |
4 64 70
|
wunxp |
⊢ ( 𝜑 → ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ∈ 𝑈 ) |
72 |
4 54
|
wunrn |
⊢ ( 𝜑 → ran ( Hom ‘ 𝑇 ) ∈ 𝑈 ) |
73 |
4 72
|
wununi |
⊢ ( 𝜑 → ∪ ran ( Hom ‘ 𝑇 ) ∈ 𝑈 ) |
74 |
4 73 73
|
wunxp |
⊢ ( 𝜑 → ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ∈ 𝑈 ) |
75 |
4 71 74
|
wunpm |
⊢ ( 𝜑 → ( ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ↑pm ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ) ∈ 𝑈 ) |
76 |
|
fvex |
⊢ ( comp ‘ 𝑋 ) ∈ V |
77 |
76
|
rnex |
⊢ ran ( comp ‘ 𝑋 ) ∈ V |
78 |
77
|
uniex |
⊢ ∪ ran ( comp ‘ 𝑋 ) ∈ V |
79 |
78
|
rnex |
⊢ ran ∪ ran ( comp ‘ 𝑋 ) ∈ V |
80 |
79
|
uniex |
⊢ ∪ ran ∪ ran ( comp ‘ 𝑋 ) ∈ V |
81 |
80
|
pwex |
⊢ 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) ∈ V |
82 |
|
fvex |
⊢ ( comp ‘ 𝑌 ) ∈ V |
83 |
82
|
rnex |
⊢ ran ( comp ‘ 𝑌 ) ∈ V |
84 |
83
|
uniex |
⊢ ∪ ran ( comp ‘ 𝑌 ) ∈ V |
85 |
84
|
rnex |
⊢ ran ∪ ran ( comp ‘ 𝑌 ) ∈ V |
86 |
85
|
uniex |
⊢ ∪ ran ∪ ran ( comp ‘ 𝑌 ) ∈ V |
87 |
86
|
pwex |
⊢ 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ∈ V |
88 |
81 87
|
xpex |
⊢ ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ∈ V |
89 |
|
fvex |
⊢ ( Hom ‘ 𝑇 ) ∈ V |
90 |
89
|
rnex |
⊢ ran ( Hom ‘ 𝑇 ) ∈ V |
91 |
90
|
uniex |
⊢ ∪ ran ( Hom ‘ 𝑇 ) ∈ V |
92 |
91 91
|
xpex |
⊢ ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ∈ V |
93 |
|
ovssunirn |
⊢ ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) ⊆ ∪ ran ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) |
94 |
|
ovssunirn |
⊢ ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ⊆ ∪ ran ( comp ‘ 𝑋 ) |
95 |
|
rnss |
⊢ ( ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ⊆ ∪ ran ( comp ‘ 𝑋 ) → ran ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ⊆ ran ∪ ran ( comp ‘ 𝑋 ) ) |
96 |
|
uniss |
⊢ ( ran ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ⊆ ran ∪ ran ( comp ‘ 𝑋 ) → ∪ ran ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ⊆ ∪ ran ∪ ran ( comp ‘ 𝑋 ) ) |
97 |
94 95 96
|
mp2b |
⊢ ∪ ran ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ⊆ ∪ ran ∪ ran ( comp ‘ 𝑋 ) |
98 |
93 97
|
sstri |
⊢ ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) ⊆ ∪ ran ∪ ran ( comp ‘ 𝑋 ) |
99 |
|
ovex |
⊢ ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) ∈ V |
100 |
99
|
elpw |
⊢ ( ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) ∈ 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) ↔ ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) ⊆ ∪ ran ∪ ran ( comp ‘ 𝑋 ) ) |
101 |
98 100
|
mpbir |
⊢ ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) ∈ 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) |
102 |
|
ovssunirn |
⊢ ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) ⊆ ∪ ran ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) |
103 |
|
ovssunirn |
⊢ ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ⊆ ∪ ran ( comp ‘ 𝑌 ) |
104 |
|
rnss |
⊢ ( ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ⊆ ∪ ran ( comp ‘ 𝑌 ) → ran ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ⊆ ran ∪ ran ( comp ‘ 𝑌 ) ) |
105 |
|
uniss |
⊢ ( ran ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ⊆ ran ∪ ran ( comp ‘ 𝑌 ) → ∪ ran ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ⊆ ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) |
106 |
103 104 105
|
mp2b |
⊢ ∪ ran ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ⊆ ∪ ran ∪ ran ( comp ‘ 𝑌 ) |
107 |
102 106
|
sstri |
⊢ ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) ⊆ ∪ ran ∪ ran ( comp ‘ 𝑌 ) |
108 |
|
ovex |
⊢ ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) ∈ V |
109 |
108
|
elpw |
⊢ ( ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) ∈ 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ↔ ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) ⊆ ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) |
110 |
107 109
|
mpbir |
⊢ ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) ∈ 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) |
111 |
|
opelxpi |
⊢ ( ( ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) ∈ 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) ∧ ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) ∈ 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) → 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ∈ ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ) |
112 |
101 110 111
|
mp2an |
⊢ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ∈ ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) |
113 |
112
|
rgen2w |
⊢ ∀ 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) ∀ 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ∈ ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) |
114 |
|
eqid |
⊢ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) = ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) |
115 |
114
|
fmpo |
⊢ ( ∀ 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) ∀ 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ∈ ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ↔ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) : ( ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) × ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ) ⟶ ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ) |
116 |
113 115
|
mpbi |
⊢ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) : ( ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) × ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ) ⟶ ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) |
117 |
|
ovssunirn |
⊢ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) ⊆ ∪ ran ( Hom ‘ 𝑇 ) |
118 |
|
fvssunirn |
⊢ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ⊆ ∪ ran ( Hom ‘ 𝑇 ) |
119 |
|
xpss12 |
⊢ ( ( ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) ⊆ ∪ ran ( Hom ‘ 𝑇 ) ∧ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ⊆ ∪ ran ( Hom ‘ 𝑇 ) ) → ( ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) × ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ) ⊆ ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ) |
120 |
117 118 119
|
mp2an |
⊢ ( ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) × ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ) ⊆ ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) |
121 |
|
elpm2r |
⊢ ( ( ( ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ∈ V ∧ ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ∈ V ) ∧ ( ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) : ( ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) × ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ) ⟶ ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ∧ ( ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) × ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ) ⊆ ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ) ) → ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ∈ ( ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ↑pm ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ) ) |
122 |
88 92 116 120 121
|
mp4an |
⊢ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ∈ ( ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ↑pm ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ) |
123 |
122
|
rgen2w |
⊢ ∀ 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) ∀ 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ∈ ( ( 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑋 ) × 𝒫 ∪ ran ∪ ran ( comp ‘ 𝑌 ) ) ↑pm ( ∪ ran ( Hom ‘ 𝑇 ) × ∪ ran ( Hom ‘ 𝑇 ) ) ) |
124 |
|
eqid |
⊢ ( 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) , 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) = ( 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) , 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) |
125 |
124
|
fmpo |
⊢ ( ∀ 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) ∀ 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 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 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 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 ) ) ) ) |
126 |
123 125
|
mpbi |
⊢ ( 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) , 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 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 ) ) ) |
127 |
126
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) , 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 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 ) ) ) ) |
128 |
4 58 75 127
|
wunf |
⊢ ( 𝜑 → ( 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) , 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) ∈ 𝑈 ) |
129 |
4 57 128
|
wunop |
⊢ ( 𝜑 → 〈 ( comp ‘ ndx ) , ( 𝑥 ∈ ( ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) × ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ) , 𝑦 ∈ ( ( Base ‘ 𝑋 ) × ( Base ‘ 𝑌 ) ) ↦ ( 𝑔 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 ∈ 𝑈 ) |
130 |
4 27 55 129
|
wuntp |
⊢ ( 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 ∈ ( ( 2nd ‘ 𝑥 ) ( Hom ‘ 𝑇 ) 𝑦 ) , 𝑓 ∈ ( ( Hom ‘ 𝑇 ) ‘ 𝑥 ) ↦ 〈 ( ( 1st ‘ 𝑔 ) ( 〈 ( 1st ‘ ( 1st ‘ 𝑥 ) ) , ( 1st ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑋 ) ( 1st ‘ 𝑦 ) ) ( 1st ‘ 𝑓 ) ) , ( ( 2nd ‘ 𝑔 ) ( 〈 ( 2nd ‘ ( 1st ‘ 𝑥 ) ) , ( 2nd ‘ ( 2nd ‘ 𝑥 ) ) 〉 ( comp ‘ 𝑌 ) ( 2nd ‘ 𝑦 ) ) ( 2nd ‘ 𝑓 ) ) 〉 ) ) 〉 } ∈ 𝑈 ) |
131 |
20 130
|
eqeltrd |
⊢ ( 𝜑 → 𝑇 ∈ 𝑈 ) |
132 |
1 2 4
|
catcbas |
⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Cat ) ) |
133 |
6 132
|
eleqtrd |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝑈 ∩ Cat ) ) |
134 |
133
|
elin2d |
⊢ ( 𝜑 → 𝑋 ∈ Cat ) |
135 |
7 132
|
eleqtrd |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑈 ∩ Cat ) ) |
136 |
135
|
elin2d |
⊢ ( 𝜑 → 𝑌 ∈ Cat ) |
137 |
3 134 136
|
xpccat |
⊢ ( 𝜑 → 𝑇 ∈ Cat ) |
138 |
131 137
|
elind |
⊢ ( 𝜑 → 𝑇 ∈ ( 𝑈 ∩ Cat ) ) |
139 |
138 132
|
eleqtrrd |
⊢ ( 𝜑 → 𝑇 ∈ 𝐵 ) |