Theorem catcxpcclOLD 17925

Description: Obsolete proof of catcxpccl 17924 as of 14-Oct-2024. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

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
catcxpcclOLD	⊢ (𝜑 → 𝑇 ∈ 𝐵)

Proof of Theorem catcxpcclOLD

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 2738	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2738	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
4		eqid 2738	. . . . 5 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
5		eqid 2738	. . . . 5 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
6		eqid 2738	. . . . 5 ⊢ (comp‘𝑋) = (comp‘𝑋)
7		eqid 2738	. . . . 5 ⊢ (comp‘𝑌) = (comp‘𝑌)
8		catcxpccl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		catcxpccl.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		eqidd 2739	. . . . 5 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
11	1, 2, 3	xpcbas 17895	. . . . . . 7 ⊢ ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12		eqid 2738	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
13	1, 11, 4, 5, 12	xpchomfval 17896	. . . . . 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 2739	. . . . 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 17894	. . . 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 16913	. . . . . . 7 ⊢ Base = Slot 1
19		catcxpccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	17, 19	wunndx 16896	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 17, 20	wunstr 16889	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22		catcxpccl.c	. . . . . . . . . . 11 ⊢ 𝐶 = (CatCat‘𝑈)
23		catcxpccl.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
24	22, 23, 17	catcbas 17816	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
25	8, 24	eleqtrd 2841	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
26	25	elin1d 4132	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
27	18, 17, 26	wunstr 16889	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
28	9, 24	eleqtrd 2841	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
29	28	elin1d 4132	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
30	18, 17, 29	wunstr 16889	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑌) ∈ 𝑈)
31	17, 27, 30	wunxp 10480	. . . . . 6 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
32	17, 21, 31	wunop 10478	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
33		df-hom 16986	. . . . . . 7 ⊢ Hom = Slot ;14
34	33, 17, 20	wunstr 16889	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
35	17, 31, 31	wunxp 10480	. . . . . . . 8 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
36	33, 17, 26	wunstr 16889	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
37	17, 36	wunrn 10485	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
38	17, 37	wununi 10462	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑋) ∈ 𝑈)
39	33, 17, 29	wunstr 16889	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
40	17, 39	wunrn 10485	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
41	17, 40	wununi 10462	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑌) ∈ 𝑈)
42	17, 38, 41	wunxp 10480	. . . . . . . . 9 ⊢ (𝜑 → (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
43	17, 42	wunpw 10463	. . . . . . . 8 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
44		ovssunirn 7311	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑋)
45		ovssunirn 7311	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑌)
46		xpss12 5604	. . . . . . . . . . . . 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)))
47	44, 45, 46	mp2an 689	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ⊆ (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
48		ovex 7308	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ∈ V
49		ovex 7308	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ∈ V
50	48, 49	xpex 7603	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ V
51	50	elpw 4537	. . . . . . . . . . . 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)))
52	47, 51	mpbir 230	. . . . . . . . . . 11 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
53	52	rgen2w 3077	. . . . . . . . . 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))
54		eqid 2738	. . . . . . . . . . 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))))
55	54	fmpo 7908	. . . . . . . . . 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)))
56	53, 55	mpbi 229	. . . . . . . . 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))
57	56	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)))
58	17, 35, 43, 57	wunf 10483	. . . . . . 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)
59	13, 58	eqeltrid 2843	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
60	17, 34, 59	wunop 10478	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
61		df-cco 16987	. . . . . . 7 ⊢ comp = Slot ;15
62	61, 17, 20	wunstr 16889	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
63	17, 35, 31	wunxp 10480	. . . . . . 7 ⊢ (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
64	61, 17, 26	wunstr 16889	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
65	17, 64	wunrn 10485	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
66	17, 65	wununi 10462	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
67	17, 66	wunrn 10485	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
68	17, 67	wununi 10462	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
69	17, 68	wunpw 10463	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
70	61, 17, 29	wunstr 16889	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
71	17, 70	wunrn 10485	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
72	17, 71	wununi 10462	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
73	17, 72	wunrn 10485	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
74	17, 73	wununi 10462	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
75	17, 74	wunpw 10463	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
76	17, 69, 75	wunxp 10480	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ 𝑈)
77	17, 59	wunrn 10485	. . . . . . . . . 10 ⊢ (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
78	17, 77	wununi 10462	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (Hom ‘𝑇) ∈ 𝑈)
79	17, 78, 78	wunxp 10480	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ 𝑈)
80	17, 76, 79	wunpm 10481	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑pm (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))) ∈ 𝑈)
81		fvex 6787	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑋) ∈ V
82	81	rnex 7759	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑋) ∈ V
83	82	uniex 7594	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑋) ∈ V
84	83	rnex 7759	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑋) ∈ V
85	84	uniex 7594	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑋) ∈ V
86	85	pwex 5303	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ V
87		fvex 6787	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑌) ∈ V
88	87	rnex 7759	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑌) ∈ V
89	88	uniex 7594	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑌) ∈ V
90	89	rnex 7759	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
91	90	uniex 7594	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
92	91	pwex 5303	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
93	86, 92	xpex 7603	. . . . . . . . . . 11 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ V
94		fvex 6787	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝑇) ∈ V
95	94	rnex 7759	. . . . . . . . . . . . 13 ⊢ ran (Hom ‘𝑇) ∈ V
96	95	uniex 7594	. . . . . . . . . . . 12 ⊢ ∪ ran (Hom ‘𝑇) ∈ V
97	96, 96	xpex 7603	. . . . . . . . . . 11 ⊢ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ V
98		ovssunirn 7311	. . . . . . . . . . . . . . . 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))
99		ovssunirn 7311	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦)) ⊆ ∪ ran (comp‘𝑋)
100		rnss 5848	. . . . . . . . . . . . . . . . 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))
101		uniss 4847	. . . . . . . . . . . . . . . . 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))
102	99, 100, 101	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦)) ⊆ ∪ ran ∪ ran (comp‘𝑋)
103	98, 102	sstri 3930	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑋)
104		ovex 7308	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ V
105	104	elpw 4537	. . . . . . . . . . . . . . 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))
106	103, 105	mpbir 230	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑋)
107		ovssunirn 7311	. . . . . . . . . . . . . . . 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))
108		ovssunirn 7311	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦)) ⊆ ∪ ran (comp‘𝑌)
109		rnss 5848	. . . . . . . . . . . . . . . . 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))
110		uniss 4847	. . . . . . . . . . . . . . . . 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))
111	108, 109, 110	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
112	107, 111	sstri 3930	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
113		ovex 7308	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ V
114	113	elpw 4537	. . . . . . . . . . . . . . 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))
115	112, 114	mpbir 230	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌)
116		opelxpi 5626	. . . . . . . . . . . . . 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)))
117	106, 115, 116	mp2an 689	. . . . . . . . . . . . 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))
118	117	rgen2w 3077	. . . . . . . . . . . 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))
119		eqid 2738	. . . . . . . . . . . . 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))⟩)
120	119	fmpo 7908	. . . . . . . . . . . 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)))
121	118, 120	mpbi 229	. . . . . . . . . . 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))
122		ovssunirn 7311	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑥)(Hom ‘𝑇)𝑦) ⊆ ∪ ran (Hom ‘𝑇)
123		fvssunirn 6803	. . . . . . . . . . . 12 ⊢ ((Hom ‘𝑇)‘𝑥) ⊆ ∪ ran (Hom ‘𝑇)
124		xpss12 5604	. . . . . . . . . . . 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)))
125	122, 123, 124	mp2an 689	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))
126		elpm2r 8633	. . . . . . . . . . 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))))
127	93, 97, 121, 125, 126	mp4an 690	. . . . . . . . . 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)))
128	127	rgen2w 3077	. . . . . . . . 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)))
129		eqid 2738	. . . . . . . . . 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))⟩))
130	129	fmpo 7908	. . . . . . . . 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))))
131	128, 130	mpbi 229	. . . . . . . 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)))
132	131	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))))
133	17, 63, 80, 132	wunf 10483	. . . . . 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)
134	17, 62, 133	wunop 10478	. . . . 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)
135	17, 32, 60, 134	wuntp 10467	. . . 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)
136	16, 135	eqeltrd 2839	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑈)
137	25	elin2d 4133	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
138	28	elin2d 4133	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Cat)
139	1, 137, 138	xpccat 17907	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
140	136, 139	elind 4128	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑈 ∩ Cat))
141	140, 24	eleqtrrd 2842	1 ⊢ (𝜑 → 𝑇 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  {ctp 4565  ⟨cop 4567  ∪ cuni 4839   × cxp 5587  ran crn 5590  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ωcom 7712  1^st c1st 7829  2^nd c2nd 7830   ↑_pm cpm 8616  WUnicwun 10456  1c1 10872  4c4 12030  5c5 12031  ;cdc 12437  ndxcnx 16894  Basecbs 16912  Hom chom 16973  compcco 16974  Catccat 17373  CatCatccatc 17813   ×_c cxpc 17885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-ec 8500  df-qs 8504  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-wun 10458  df-ni 10628  df-pli 10629  df-mi 10630  df-lti 10631  df-plpq 10664  df-mpq 10665  df-ltpq 10666  df-enq 10667  df-nq 10668  df-erq 10669  df-plq 10670  df-mq 10671  df-1nq 10672  df-rq 10673  df-ltnq 10674  df-np 10737  df-plp 10739  df-ltp 10741  df-enr 10811  df-nr 10812  df-c 10877  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-slot 16883  df-ndx 16895  df-base 16913  df-hom 16986  df-cco 16987  df-cat 17377  df-cid 17378  df-catc 17814  df-xpc 17889

This theorem is referenced by: (None)