Theorem catcxpcclOLD 17669

Description: Obsolete proof of catcxpccl 17668 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 2736	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2736	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
4		eqid 2736	. . . . 5 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
5		eqid 2736	. . . . 5 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
6		eqid 2736	. . . . 5 ⊢ (comp‘𝑋) = (comp‘𝑋)
7		eqid 2736	. . . . 5 ⊢ (comp‘𝑌) = (comp‘𝑌)
8		catcxpccl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		catcxpccl.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		eqidd 2737	. . . . 5 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
11	1, 2, 3	xpcbas 17639	. . . . . . 7 ⊢ ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12		eqid 2736	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
13	1, 11, 4, 5, 12	xpchomfval 17640	. . . . . 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 2737	. . . . 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 17638	. . . 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 16672	. . . . . . 7 ⊢ Base = Slot 1
19		catcxpccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	17, 19	wunndx 16686	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 17, 20	wunstr 16689	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22		catcxpccl.c	. . . . . . . . . . 11 ⊢ 𝐶 = (CatCat‘𝑈)
23		catcxpccl.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝐶)
24	22, 23, 17	catcbas 17561	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
25	8, 24	eleqtrd 2833	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
26	25	elin1d 4098	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑈)
27	18, 17, 26	wunstr 16689	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
28	9, 24	eleqtrd 2833	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
29	28	elin1d 4098	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑈)
30	18, 17, 29	wunstr 16689	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑌) ∈ 𝑈)
31	17, 27, 30	wunxp 10303	. . . . . 6 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
32	17, 21, 31	wunop 10301	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
33		df-hom 16773	. . . . . . 7 ⊢ Hom = Slot ;14
34	33, 17, 20	wunstr 16689	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
35	17, 31, 31	wunxp 10303	. . . . . . . 8 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
36	33, 17, 26	wunstr 16689	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
37	17, 36	wunrn 10308	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
38	17, 37	wununi 10285	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑋) ∈ 𝑈)
39	33, 17, 29	wunstr 16689	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
40	17, 39	wunrn 10308	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
41	17, 40	wununi 10285	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑌) ∈ 𝑈)
42	17, 38, 41	wunxp 10303	. . . . . . . . 9 ⊢ (𝜑 → (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
43	17, 42	wunpw 10286	. . . . . . . 8 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
44		ovssunirn 7227	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑋)
45		ovssunirn 7227	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑌)
46		xpss12 5551	. . . . . . . . . . . . 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 692	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ⊆ (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
48		ovex 7224	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ∈ V
49		ovex 7224	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ∈ V
50	48, 49	xpex 7516	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ V
51	50	elpw 4503	. . . . . . . . . . . 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 234	. . . . . . . . . . 11 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
53	52	rgen2w 3064	. . . . . . . . . 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 2736	. . . . . . . . . . 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 7816	. . . . . . . . . 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 233	. . . . . . . . 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 10306	. . . . . . 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 2835	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
60	17, 34, 59	wunop 10301	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
61		df-cco 16774	. . . . . . 7 ⊢ comp = Slot ;15
62	61, 17, 20	wunstr 16689	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
63	17, 35, 31	wunxp 10303	. . . . . . 7 ⊢ (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
64	61, 17, 26	wunstr 16689	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
65	17, 64	wunrn 10308	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
66	17, 65	wununi 10285	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
67	17, 66	wunrn 10308	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
68	17, 67	wununi 10285	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
69	17, 68	wunpw 10286	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
70	61, 17, 29	wunstr 16689	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
71	17, 70	wunrn 10308	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
72	17, 71	wununi 10285	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
73	17, 72	wunrn 10308	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
74	17, 73	wununi 10285	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
75	17, 74	wunpw 10286	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
76	17, 69, 75	wunxp 10303	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ 𝑈)
77	17, 59	wunrn 10308	. . . . . . . . . 10 ⊢ (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
78	17, 77	wununi 10285	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (Hom ‘𝑇) ∈ 𝑈)
79	17, 78, 78	wunxp 10303	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ 𝑈)
80	17, 76, 79	wunpm 10304	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑pm (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))) ∈ 𝑈)
81		fvex 6708	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑋) ∈ V
82	81	rnex 7668	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑋) ∈ V
83	82	uniex 7507	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑋) ∈ V
84	83	rnex 7668	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑋) ∈ V
85	84	uniex 7507	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑋) ∈ V
86	85	pwex 5258	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ V
87		fvex 6708	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑌) ∈ V
88	87	rnex 7668	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑌) ∈ V
89	88	uniex 7507	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑌) ∈ V
90	89	rnex 7668	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
91	90	uniex 7507	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
92	91	pwex 5258	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
93	86, 92	xpex 7516	. . . . . . . . . . 11 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ V
94		fvex 6708	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝑇) ∈ V
95	94	rnex 7668	. . . . . . . . . . . . 13 ⊢ ran (Hom ‘𝑇) ∈ V
96	95	uniex 7507	. . . . . . . . . . . 12 ⊢ ∪ ran (Hom ‘𝑇) ∈ V
97	96, 96	xpex 7516	. . . . . . . . . . 11 ⊢ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ V
98		ovssunirn 7227	. . . . . . . . . . . . . . . 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 7227	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦)) ⊆ ∪ ran (comp‘𝑋)
100		rnss 5793	. . . . . . . . . . . . . . . . 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 4813	. . . . . . . . . . . . . . . . 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 3896	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑋)
104		ovex 7224	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ V
105	104	elpw 4503	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑋)
107		ovssunirn 7227	. . . . . . . . . . . . . . . 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 7227	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦)) ⊆ ∪ ran (comp‘𝑌)
109		rnss 5793	. . . . . . . . . . . . . . . . 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 4813	. . . . . . . . . . . . . . . . 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 3896	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
113		ovex 7224	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ V
114	113	elpw 4503	. . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌)
116		opelxpi 5573	. . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 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 3064	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 7816	. . . . . . . . . . . 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 233	. . . . . . . . . . 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 7227	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑥)(Hom ‘𝑇)𝑦) ⊆ ∪ ran (Hom ‘𝑇)
123		fvssunirn 6724	. . . . . . . . . . . 12 ⊢ ((Hom ‘𝑇)‘𝑥) ⊆ ∪ ran (Hom ‘𝑇)
124		xpss12 5551	. . . . . . . . . . . 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 692	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))
126		elpm2r 8504	. . . . . . . . . . 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 693	. . . . . . . . . 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 3064	. . . . . . . . 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 2736	. . . . . . . . . 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 7816	. . . . . . . . 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 233	. . . . . . . 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 10306	. . . . . 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 10301	. . . . 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 10290	. . . 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 2831	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑈)
137	25	elin2d 4099	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
138	28	elin2d 4099	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Cat)
139	1, 137, 138	xpccat 17651	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
140	136, 139	elind 4094	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑈 ∩ Cat))
141	140, 24	eleqtrrd 2834	1 ⊢ (𝜑 → 𝑇 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112  ∀wral 3051  Vcvv 3398   ∩ cin 3852   ⊆ wss 3853  𝒫 cpw 4499  {ctp 4531  ⟨cop 4533  ∪ cuni 4805   × cxp 5534  ran crn 5537  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191   ∈ cmpo 7193  ωcom 7622  1^st c1st 7737  2^nd c2nd 7738   ↑_pm cpm 8487  WUnicwun 10279  1c1 10695  4c4 11852  5c5 11853  ;cdc 12258  ndxcnx 16663  Basecbs 16666  Hom chom 16760  compcco 16761  Catccat 17121  CatCatccatc 17558   ×_c cxpc 17629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-inf2 9234  ax-cnex 10750  ax-resscn 10751  ax-1cn 10752  ax-icn 10753  ax-addcl 10754  ax-addrcl 10755  ax-mulcl 10756  ax-mulrcl 10757  ax-mulcom 10758  ax-addass 10759  ax-mulass 10760  ax-distr 10761  ax-i2m1 10762  ax-1ne0 10763  ax-1rid 10764  ax-rnegex 10765  ax-rrecex 10766  ax-cnre 10767  ax-pre-lttri 10768  ax-pre-lttrn 10769  ax-pre-ltadd 10770  ax-pre-mulgt0 10771

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-oadd 8184  df-omul 8185  df-er 8369  df-ec 8371  df-qs 8375  df-map 8488  df-pm 8489  df-en 8605  df-dom 8606  df-sdom 8607  df-fin 8608  df-wun 10281  df-ni 10451  df-pli 10452  df-mi 10453  df-lti 10454  df-plpq 10487  df-mpq 10488  df-ltpq 10489  df-enq 10490  df-nq 10491  df-erq 10492  df-plq 10493  df-mq 10494  df-1nq 10495  df-rq 10496  df-ltnq 10497  df-np 10560  df-plp 10562  df-ltp 10564  df-enr 10634  df-nr 10635  df-c 10700  df-pnf 10834  df-mnf 10835  df-xr 10836  df-ltxr 10837  df-le 10838  df-sub 11029  df-neg 11030  df-nn 11796  df-2 11858  df-3 11859  df-4 11860  df-5 11861  df-6 11862  df-7 11863  df-8 11864  df-9 11865  df-n0 12056  df-z 12142  df-dec 12259  df-uz 12404  df-fz 13061  df-struct 16668  df-ndx 16669  df-slot 16670  df-base 16672  df-hom 16773  df-cco 16774  df-cat 17125  df-cid 17126  df-catc 17559  df-xpc 17633

This theorem is referenced by: (None)