Theorem catcxpccl 16847

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 2622	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2622	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
4		eqid 2622	. . . . 5 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
5		eqid 2622	. . . . 5 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
6		eqid 2622	. . . . 5 ⊢ (comp‘𝑋) = (comp‘𝑋)
7		eqid 2622	. . . . 5 ⊢ (comp‘𝑌) = (comp‘𝑌)
8		catcxpccl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		catcxpccl.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		eqidd 2623	. . . . 5 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
11	1, 2, 3	xpcbas 16818	. . . . . . 7 ⊢ ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12		eqid 2622	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
13	1, 11, 4, 5, 12	xpchomfval 16819	. . . . . 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 2623	. . . . 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 16817	. . . 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 15863	. . . . . . 7 ⊢ Base = Slot 1
19		catcxpccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	17, 19	wunndx 15878	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 17, 20	wunstr 15881	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22		inss1 3833	. . . . . . . . 9 ⊢ (𝑈 ∩ Cat) ⊆ 𝑈
23		catcxpccl.c	. . . . . . . . . . 11 ⊢ 𝐶 = (CatCat‘𝑈)
24		catcxpccl.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
25	23, 24, 17	catcbas 16747	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
26	8, 25	eleqtrd 2703	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
27	22, 26	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
28	18, 17, 27	wunstr 15881	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
29	9, 25	eleqtrd 2703	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
30	22, 29	sseldi 3601	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
31	18, 17, 30	wunstr 15881	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑌) ∈ 𝑈)
32	17, 28, 31	wunxp 9546	. . . . . 6 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
33	17, 21, 32	wunop 9544	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
34		df-hom 15966	. . . . . . 7 ⊢ Hom = Slot ;14
35	34, 17, 20	wunstr 15881	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
36	17, 32, 32	wunxp 9546	. . . . . . . 8 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
37	34, 17, 27	wunstr 15881	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
38	17, 37	wunrn 9551	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
39	17, 38	wununi 9528	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑋) ∈ 𝑈)
40	34, 17, 30	wunstr 15881	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
41	17, 40	wunrn 9551	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
42	17, 41	wununi 9528	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑌) ∈ 𝑈)
43	17, 39, 42	wunxp 9546	. . . . . . . . 9 ⊢ (𝜑 → (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
44	17, 43	wunpw 9529	. . . . . . . 8 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
45		ovssunirn 6681	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑋)
46		ovssunirn 6681	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑌)
47		xpss12 5225	. . . . . . . . . . . . 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 708	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ⊆ (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
49		ovex 6678	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ∈ V
50		ovex 6678	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ∈ V
51	49, 50	xpex 6962	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ V
52	51	elpw 4164	. . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
54	53	rgen2w 2925	. . . . . . . . . 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 2622	. . . . . . . . . . 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 7237	. . . . . . . . . 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 220	. . . . . . . . 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 9549	. . . . . . 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 2705	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
61	17, 35, 60	wunop 9544	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
62		df-cco 15967	. . . . . . 7 ⊢ comp = Slot ;15
63	62, 17, 20	wunstr 15881	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
64	17, 36, 32	wunxp 9546	. . . . . . 7 ⊢ (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
65	62, 17, 27	wunstr 15881	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
66	17, 65	wunrn 9551	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
67	17, 66	wununi 9528	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
68	17, 67	wunrn 9551	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
69	17, 68	wununi 9528	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
70	17, 69	wunpw 9529	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
71	62, 17, 30	wunstr 15881	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
72	17, 71	wunrn 9551	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
73	17, 72	wununi 9528	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
74	17, 73	wunrn 9551	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
75	17, 74	wununi 9528	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
76	17, 75	wunpw 9529	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
77	17, 70, 76	wunxp 9546	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ 𝑈)
78	17, 60	wunrn 9551	. . . . . . . . . 10 ⊢ (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
79	17, 78	wununi 9528	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (Hom ‘𝑇) ∈ 𝑈)
80	17, 79, 79	wunxp 9546	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ 𝑈)
81	17, 77, 80	wunpm 9547	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑pm (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))) ∈ 𝑈)
82		fvex 6201	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑋) ∈ V
83	82	rnex 7100	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑋) ∈ V
84	83	uniex 6953	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑋) ∈ V
85	84	rnex 7100	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑋) ∈ V
86	85	uniex 6953	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑋) ∈ V
87	86	pwex 4848	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ V
88		fvex 6201	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑌) ∈ V
89	88	rnex 7100	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑌) ∈ V
90	89	uniex 6953	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑌) ∈ V
91	90	rnex 7100	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
92	91	uniex 6953	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
93	92	pwex 4848	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
94	87, 93	xpex 6962	. . . . . . . . . . 11 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ V
95		fvex 6201	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝑇) ∈ V
96	95	rnex 7100	. . . . . . . . . . . . 13 ⊢ ran (Hom ‘𝑇) ∈ V
97	96	uniex 6953	. . . . . . . . . . . 12 ⊢ ∪ ran (Hom ‘𝑇) ∈ V
98	97, 97	xpex 6962	. . . . . . . . . . 11 ⊢ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ V
99		ovssunirn 6681	. . . . . . . . . . . . . . . 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 6681	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦)) ⊆ ∪ ran (comp‘𝑋)
101		rnss 5354	. . . . . . . . . . . . . . . . 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 4458	. . . . . . . . . . . . . . . . 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 3612	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑋)
105		ovex 6678	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ V
106	105	elpw 4164	. . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑋)
108		ovssunirn 6681	. . . . . . . . . . . . . . . 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 6681	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦)) ⊆ ∪ ran (comp‘𝑌)
110		rnss 5354	. . . . . . . . . . . . . . . . 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 4458	. . . . . . . . . . . . . . . . 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 3612	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
114		ovex 6678	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ V
115	114	elpw 4164	. . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌)
117		opelxpi 5148	. . . . . . . . . . . . . 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 708	. . . . . . . . . . . . 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 2925	. . . . . . . . . . . 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 2622	. . . . . . . . . . . . 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 7237	. . . . . . . . . . . 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 220	. . . . . . . . . . 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 6681	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑥)(Hom ‘𝑇)𝑦) ⊆ ∪ ran (Hom ‘𝑇)
124		fvssunirn 6217	. . . . . . . . . . . 12 ⊢ ((Hom ‘𝑇)‘𝑥) ⊆ ∪ ran (Hom ‘𝑇)
125		xpss12 5225	. . . . . . . . . . . 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 708	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))
127		elpm2r 7875	. . . . . . . . . . 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 709	. . . . . . . . . 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 2925	. . . . . . . . 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 2622	. . . . . . . . . 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 7237	. . . . . . . . 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 220	. . . . . . . 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 9549	. . . . . 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 9544	. . . . 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 9533	. . . 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 2701	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑈)
138		inss2 3834	. . . . 5 ⊢ (𝑈 ∩ Cat) ⊆ Cat
139	138, 26	sseldi 3601	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
140	138, 29	sseldi 3601	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Cat)
141	1, 139, 140	xpccat 16830	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
142	137, 141	elind 3798	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑈 ∩ Cat))
143	142, 25	eleqtrrd 2704	1 ⊢ (𝜑 → 𝑇 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  𝒫 cpw 4158  {ctp 4181  ⟨cop 4183  ∪ cuni 4436   × cxp 5112  ran crn 5115  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065  1^st c1st 7166  2^nd c2nd 7167   ↑_pm cpm 7858  WUnicwun 9522  1c1 9937  4c4 11072  5c5 11073  ;cdc 11493  ndxcnx 15854  Basecbs 15857  Hom chom 15952  compcco 15953  Catccat 16325  CatCatccatc 16744   ×_c cxpc 16808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wun 9524  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807  df-enr 9877  df-nr 9878  df-c 9942  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-catc 16745  df-xpc 16812

This theorem is referenced by: (None)