Theorem catcxpccl 16670

Description: The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
catcxpccl.c	⊢ 𝐶 = (CatCat‘𝑈)
catcxpccl.b	⊢ 𝐵 = (Base‘𝐶)
catcxpccl.o	⊢ 𝑇 = (𝑋 ×c 𝑌)
catcxpccl.u	⊢ (𝜑 → 𝑈 ∈ WUni)
catcxpccl.1	⊢ (𝜑 → ω ∈ 𝑈)
catcxpccl.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
catcxpccl.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
catcxpccl	⊢ (𝜑 → 𝑇 ∈ 𝐵)

Proof of Theorem catcxpccl

Dummy variables 𝑓 𝑔 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catcxpccl.o	. . . . 5 ⊢ 𝑇 = (𝑋 ×c 𝑌)
2		eqid 2610	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2610	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
4		eqid 2610	. . . . 5 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
5		eqid 2610	. . . . 5 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
6		eqid 2610	. . . . 5 ⊢ (comp‘𝑋) = (comp‘𝑋)
7		eqid 2610	. . . . 5 ⊢ (comp‘𝑌) = (comp‘𝑌)
8		catcxpccl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		catcxpccl.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		eqidd 2611	. . . . 5 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
11	1, 2, 3	xpcbas 16641	. . . . . . 7 ⊢ ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12		eqid 2610	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
13	1, 11, 4, 5, 12	xpchomfval 16642	. . . . . 6 ⊢ (Hom ‘𝑇) = (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))))
14	13	a1i 11	. . . . 5 ⊢ (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)))))
15		eqidd 2611	. . . . 5 ⊢ (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))⟩)))
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15	xpcval 16640	. . . 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 ∈ ((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))⟩))⟩})
17		catcxpccl.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni)
18		df-base 15700	. . . . . . 7 ⊢ Base = Slot 1
19		catcxpccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	17, 19	wunndx 15711	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 17, 20	wunstr 15714	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22		inss1 3795	. . . . . . . . 9 ⊢ (𝑈 ∩ Cat) ⊆ 𝑈
23		catcxpccl.c	. . . . . . . . . . 11 ⊢ 𝐶 = (CatCat‘𝑈)
24		catcxpccl.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
25	23, 24, 17	catcbas 16570	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
26	8, 25	eleqtrd 2690	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
27	22, 26	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
28	18, 17, 27	wunstr 15714	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
29	9, 25	eleqtrd 2690	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
30	22, 29	sseldi 3566	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
31	18, 17, 30	wunstr 15714	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑌) ∈ 𝑈)
32	17, 28, 31	wunxp 9425	. . . . . 6 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
33	17, 21, 32	wunop 9423	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
34		df-hom 15793	. . . . . . 7 ⊢ Hom = Slot ;14
35	34, 17, 20	wunstr 15714	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
36	17, 32, 32	wunxp 9425	. . . . . . . 8 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
37	34, 17, 27	wunstr 15714	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
38	17, 37	wunrn 9430	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
39	17, 38	wununi 9407	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑋) ∈ 𝑈)
40	34, 17, 30	wunstr 15714	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
41	17, 40	wunrn 9430	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
42	17, 41	wununi 9407	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑌) ∈ 𝑈)
43	17, 39, 42	wunxp 9425	. . . . . . . . 9 ⊢ (𝜑 → (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
44	17, 43	wunpw 9408	. . . . . . . 8 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
45		ovssunirn 6579	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑋)
46		ovssunirn 6579	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑌)
47		xpss12 5148	. . . . . . . . . . . . 13 ⊢ ((((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)))
48	45, 46, 47	mp2an 704	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ⊆ (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
49		ovex 6577	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ∈ V
50		ovex 6577	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ∈ V
51	49, 50	xpex 6860	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ V
52	51	elpw 4114	. . . . . . . . . . . 12 ⊢ ((((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)))
53	48, 52	mpbir 220	. . . . . . . . . . 11 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
54	53	rgen2w 2909	. . . . . . . . . 10 ⊢ ∀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))
55		eqid 2610	. . . . . . . . . . 11 ⊢ (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))))
56	55	fmpt2 7126	. . . . . . . . . 10 ⊢ (∀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)))
57	54, 56	mpbi 219	. . . . . . . . 9 ⊢ (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))
58	57	a1i 11	. . . . . . . 8 ⊢ (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)))
59	17, 36, 44, 58	wunf 9428	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ ((Base‘𝑋) × (Base‘𝑌)), 𝑣 ∈ ((Base‘𝑋) × (Base‘𝑌)) ↦ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)))) ∈ 𝑈)
60	13, 59	syl5eqel 2692	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
61	17, 35, 60	wunop 9423	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
62		df-cco 15794	. . . . . . 7 ⊢ comp = Slot ;15
63	62, 17, 20	wunstr 15714	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
64	17, 36, 32	wunxp 9425	. . . . . . 7 ⊢ (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
65	62, 17, 27	wunstr 15714	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
66	17, 65	wunrn 9430	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
67	17, 66	wununi 9407	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
68	17, 67	wunrn 9430	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
69	17, 68	wununi 9407	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
70	17, 69	wunpw 9408	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
71	62, 17, 30	wunstr 15714	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
72	17, 71	wunrn 9430	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
73	17, 72	wununi 9407	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
74	17, 73	wunrn 9430	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
75	17, 74	wununi 9407	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
76	17, 75	wunpw 9408	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
77	17, 70, 76	wunxp 9425	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ 𝑈)
78	17, 60	wunrn 9430	. . . . . . . . . 10 ⊢ (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
79	17, 78	wununi 9407	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (Hom ‘𝑇) ∈ 𝑈)
80	17, 79, 79	wunxp 9425	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ 𝑈)
81	17, 77, 80	wunpm 9426	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑pm (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))) ∈ 𝑈)
82		fvex 6113	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑋) ∈ V
83	82	rnex 6992	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑋) ∈ V
84	83	uniex 6851	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑋) ∈ V
85	84	rnex 6992	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑋) ∈ V
86	85	uniex 6851	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑋) ∈ V
87	86	pwex 4774	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ V
88		fvex 6113	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑌) ∈ V
89	88	rnex 6992	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑌) ∈ V
90	89	uniex 6851	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑌) ∈ V
91	90	rnex 6992	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
92	91	uniex 6851	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
93	92	pwex 4774	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
94	87, 93	xpex 6860	. . . . . . . . . . 11 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ V
95		fvex 6113	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝑇) ∈ V
96	95	rnex 6992	. . . . . . . . . . . . 13 ⊢ ran (Hom ‘𝑇) ∈ V
97	96	uniex 6851	. . . . . . . . . . . 12 ⊢ ∪ ran (Hom ‘𝑇) ∈ V
98	97, 97	xpex 6860	. . . . . . . . . . 11 ⊢ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ V
99		ovssunirn 6579	. . . . . . . . . . . . . . . 16 ⊢ ((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))
100		ovssunirn 6579	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦)) ⊆ ∪ ran (comp‘𝑋)
101		rnss 5275	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(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))
102		uniss 4394	. . . . . . . . . . . . . . . . 17 ⊢ (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))
103	100, 101, 102	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦)) ⊆ ∪ ran ∪ ran (comp‘𝑋)
104	99, 103	sstri 3577	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑋)
105		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ V
106	105	elpw 4114	. . . . . . . . . . . . . . 15 ⊢ (((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))
107	104, 106	mpbir 220	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑋)
108		ovssunirn 6579	. . . . . . . . . . . . . . . 16 ⊢ ((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))
109		ovssunirn 6579	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦)) ⊆ ∪ ran (comp‘𝑌)
110		rnss 5275	. . . . . . . . . . . . . . . . 17 ⊢ ((⟨(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))
111		uniss 4394	. . . . . . . . . . . . . . . . 17 ⊢ (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))
112	109, 110, 111	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
113	108, 112	sstri 3577	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
114		ovex 6577	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ V
115	114	elpw 4114	. . . . . . . . . . . . . . 15 ⊢ (((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))
116	113, 115	mpbir 220	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌)
117		opelxpi 5072	. . . . . . . . . . . . . 14 ⊢ ((((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)))
118	107, 116, 117	mp2an 704	. . . . . . . . . . . . 13 ⊢ ⟨((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))
119	118	rgen2w 2909	. . . . . . . . . . . 12 ⊢ ∀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))
120		eqid 2610	. . . . . . . . . . . . 13 ⊢ (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))⟩)
121	120	fmpt2 7126	. . . . . . . . . . . 12 ⊢ (∀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)))
122	119, 121	mpbi 219	. . . . . . . . . . 11 ⊢ (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))
123		ovssunirn 6579	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑥)(Hom ‘𝑇)𝑦) ⊆ ∪ ran (Hom ‘𝑇)
124		fvssunirn 6127	. . . . . . . . . . . 12 ⊢ ((Hom ‘𝑇)‘𝑥) ⊆ ∪ ran (Hom ‘𝑇)
125		xpss12 5148	. . . . . . . . . . . 12 ⊢ ((((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)))
126	123, 124, 125	mp2an 704	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))
127		elpm2r 7761	. . . . . . . . . . 11 ⊢ ((((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))))
128	94, 98, 122, 126, 127	mp4an 705	. . . . . . . . . 10 ⊢ (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)))
129	128	rgen2w 2909	. . . . . . . . 9 ⊢ ∀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)))
130		eqid 2610	. . . . . . . . . 10 ⊢ (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))⟩))
131	130	fmpt2 7126	. . . . . . . . 9 ⊢ (∀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))))
132	129, 131	mpbi 219	. . . . . . . 8 ⊢ (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)))
133	132	a1i 11	. . . . . . 7 ⊢ (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))))
134	17, 64, 81, 133	wunf 9428	. . . . . 6 ⊢ (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)
135	17, 63, 134	wunop 9423	. . . . 5 ⊢ (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)
136	17, 33, 61, 135	wuntp 9412	. . . 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 ∈ ((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)
137	16, 136	eqeltrd 2688	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑈)
138		inss2 3796	. . . . 5 ⊢ (𝑈 ∩ Cat) ⊆ Cat
139	138, 26	sseldi 3566	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
140	138, 29	sseldi 3566	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Cat)
141	1, 139, 140	xpccat 16653	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
142	137, 141	elind 3760	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑈 ∩ Cat))
143	142, 25	eleqtrrd 2691	1 ⊢ (𝜑 → 𝑇 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {ctp 4129  ⟨cop 4131  ∪ cuni 4372   × cxp 5036  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957  1^st c1st 7057  2^nd c2nd 7058   ↑_pm cpm 7745  WUnicwun 9401  1c1 9816  4c4 10949  5c5 10950  ;cdc 11369  ndxcnx 15692  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148  CatCatccatc 16567   ×_c cxpc 16631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-wun 9403  df-ni 9573  df-pli 9574  df-mi 9575  df-lti 9576  df-plpq 9609  df-mpq 9610  df-ltpq 9611  df-enq 9612  df-nq 9613  df-erq 9614  df-plq 9615  df-mq 9616  df-1nq 9617  df-rq 9618  df-ltnq 9619  df-np 9682  df-plp 9684  df-ltp 9686  df-enr 9756  df-nr 9757  df-c 9821  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-hom 15793  df-cco 15794  df-cat 16152  df-cid 16153  df-catc 16568  df-xpc 16635

This theorem is referenced by: (None)