Theorem catcxpccl 18374

Description: The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.) (Proof shortened by AV, 14-Oct-2024.)

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 2761	. . . . 5 ⊢ (Base‘𝑋) = (Base‘𝑋)
3		eqid 2761	. . . . 5 ⊢ (Base‘𝑌) = (Base‘𝑌)
4		eqid 2761	. . . . 5 ⊢ (Hom ‘𝑋) = (Hom ‘𝑋)
5		eqid 2761	. . . . 5 ⊢ (Hom ‘𝑌) = (Hom ‘𝑌)
6		eqid 2761	. . . . 5 ⊢ (comp‘𝑋) = (comp‘𝑋)
7		eqid 2761	. . . . 5 ⊢ (comp‘𝑌) = (comp‘𝑌)
8		catcxpccl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		catcxpccl.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		eqidd 2762	. . . . 5 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) = ((Base‘𝑋) × (Base‘𝑌)))
11	1, 2, 3	xpcbas 18345	. . . . . . 7 ⊢ ((Base‘𝑋) × (Base‘𝑌)) = (Base‘𝑇)
12		eqid 2761	. . . . . . 7 ⊢ (Hom ‘𝑇) = (Hom ‘𝑇)
13	1, 11, 4, 5, 12	xpchomfval 18346	. . . . . 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 2762	. . . . 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 18344	. . . 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		baseid 17383	. . . . . . 7 ⊢ Base = Slot (Base‘ndx)
19		catcxpccl.1	. . . . . . . 8 ⊢ (𝜑 → ω ∈ 𝑈)
20	17, 19	wunndx 17366	. . . . . . 7 ⊢ (𝜑 → ndx ∈ 𝑈)
21	18, 17, 20	wunstr 17359	. . . . . 6 ⊢ (𝜑 → (Base‘ndx) ∈ 𝑈)
22		catcxpccl.c	. . . . . . . 8 ⊢ 𝐶 = (CatCat‘𝑈)
23		catcxpccl.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶)
24	22, 23, 17, 8	catcbaselcl 18282	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑋) ∈ 𝑈)
25	22, 23, 17, 9	catcbaselcl 18282	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑌) ∈ 𝑈)
26	17, 24, 25	wunxp 10802	. . . . . 6 ⊢ (𝜑 → ((Base‘𝑋) × (Base‘𝑌)) ∈ 𝑈)
27	17, 21, 26	wunop 10800	. . . . 5 ⊢ (𝜑 → ⟨(Base‘ndx), ((Base‘𝑋) × (Base‘𝑌))⟩ ∈ 𝑈)
28		homid 17576	. . . . . . 7 ⊢ Hom = Slot (Hom ‘ndx)
29	28, 17, 20	wunstr 17359	. . . . . 6 ⊢ (𝜑 → (Hom ‘ndx) ∈ 𝑈)
30	17, 26, 26	wunxp 10802	. . . . . . . 8 ⊢ (𝜑 → (((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
31	22, 23, 17, 8	catchomcl 18283	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑋) ∈ 𝑈)
32	17, 31	wunrn 10807	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑋) ∈ 𝑈)
33	17, 32	wununi 10784	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑋) ∈ 𝑈)
34	22, 23, 17, 9	catchomcl 18283	. . . . . . . . . . . 12 ⊢ (𝜑 → (Hom ‘𝑌) ∈ 𝑈)
35	17, 34	wunrn 10807	. . . . . . . . . . 11 ⊢ (𝜑 → ran (Hom ‘𝑌) ∈ 𝑈)
36	17, 35	wununi 10784	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran (Hom ‘𝑌) ∈ 𝑈)
37	17, 33, 36	wunxp 10802	. . . . . . . . 9 ⊢ (𝜑 → (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
38	17, 37	wunpw 10785	. . . . . . . 8 ⊢ (𝜑 → 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌)) ∈ 𝑈)
39		ovssunirn 7454	. . . . . . . . . . . . 13 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑋)
40		ovssunirn 7454	. . . . . . . . . . . . 13 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ⊆ ∪ ran (Hom ‘𝑌)
41		xpss12 5666	. . . . . . . . . . . . 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)))
42	39, 40, 41	mp2an 705	. . . . . . . . . . . 12 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ⊆ (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
43		ovex 7451	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) ∈ V
44		ovex 7451	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣)) ∈ V
45	43, 44	xpex 7765	. . . . . . . . . . . . 13 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ V
46	45	elpw 4561	. . . . . . . . . . . 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)))
47	42, 46	mpbir 234	. . . . . . . . . . 11 ⊢ (((1st ‘𝑢)(Hom ‘𝑋)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝑌)(2nd ‘𝑣))) ∈ 𝒫 (∪ ran (Hom ‘𝑋) × ∪ ran (Hom ‘𝑌))
48	47	rgen2w 3082	. . . . . . . . . 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))
49		eqid 2761	. . . . . . . . . . 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))))
50	49	fmpo 8077	. . . . . . . . . 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)))
51	48, 50	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))
52	51	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)))
53	17, 30, 38, 52	wunf 10805	. . . . . . 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)
54	13, 53	eqeltrid 2865	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝑇) ∈ 𝑈)
55	17, 29, 54	wunop 10800	. . . . 5 ⊢ (𝜑 → ⟨(Hom ‘ndx), (Hom ‘𝑇)⟩ ∈ 𝑈)
56		ccoid 17578	. . . . . . 7 ⊢ comp = Slot (comp‘ndx)
57	56, 17, 20	wunstr 17359	. . . . . 6 ⊢ (𝜑 → (comp‘ndx) ∈ 𝑈)
58	17, 30, 26	wunxp 10802	. . . . . . 7 ⊢ (𝜑 → ((((Base‘𝑋) × (Base‘𝑌)) × ((Base‘𝑋) × (Base‘𝑌))) × ((Base‘𝑋) × (Base‘𝑌))) ∈ 𝑈)
59	22, 23, 17, 8	catcccocl 18284	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑋) ∈ 𝑈)
60	17, 59	wunrn 10807	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑋) ∈ 𝑈)
61	17, 60	wununi 10784	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑋) ∈ 𝑈)
62	17, 61	wunrn 10807	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑋) ∈ 𝑈)
63	17, 62	wununi 10784	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
64	17, 63	wunpw 10785	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ 𝑈)
65	22, 23, 17, 9	catcccocl 18284	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (comp‘𝑌) ∈ 𝑈)
66	17, 65	wunrn 10807	. . . . . . . . . . . . 13 ⊢ (𝜑 → ran (comp‘𝑌) ∈ 𝑈)
67	17, 66	wununi 10784	. . . . . . . . . . . 12 ⊢ (𝜑 → ∪ ran (comp‘𝑌) ∈ 𝑈)
68	17, 67	wunrn 10807	. . . . . . . . . . 11 ⊢ (𝜑 → ran ∪ ran (comp‘𝑌) ∈ 𝑈)
69	17, 68	wununi 10784	. . . . . . . . . 10 ⊢ (𝜑 → ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
70	17, 69	wunpw 10785	. . . . . . . . 9 ⊢ (𝜑 → 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ 𝑈)
71	17, 64, 70	wunxp 10802	. . . . . . . 8 ⊢ (𝜑 → (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ 𝑈)
72	17, 54	wunrn 10807	. . . . . . . . . 10 ⊢ (𝜑 → ran (Hom ‘𝑇) ∈ 𝑈)
73	17, 72	wununi 10784	. . . . . . . . 9 ⊢ (𝜑 → ∪ ran (Hom ‘𝑇) ∈ 𝑈)
74	17, 73, 73	wunxp 10802	. . . . . . . 8 ⊢ (𝜑 → (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ 𝑈)
75	17, 71, 74	wunpm 10803	. . . . . . 7 ⊢ (𝜑 → ((𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ↑pm (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))) ∈ 𝑈)
76		fvex 6896	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑋) ∈ V
77	76	rnex 7920	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑋) ∈ V
78	77	uniex 7756	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑋) ∈ V
79	78	rnex 7920	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑋) ∈ V
80	79	uniex 7756	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑋) ∈ V
81	80	pwex 5342	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑋) ∈ V
82		fvex 6896	. . . . . . . . . . . . . . . . 17 ⊢ (comp‘𝑌) ∈ V
83	82	rnex 7920	. . . . . . . . . . . . . . . 16 ⊢ ran (comp‘𝑌) ∈ V
84	83	uniex 7756	. . . . . . . . . . . . . . 15 ⊢ ∪ ran (comp‘𝑌) ∈ V
85	84	rnex 7920	. . . . . . . . . . . . . 14 ⊢ ran ∪ ran (comp‘𝑌) ∈ V
86	85	uniex 7756	. . . . . . . . . . . . 13 ⊢ ∪ ran ∪ ran (comp‘𝑌) ∈ V
87	86	pwex 5342	. . . . . . . . . . . 12 ⊢ 𝒫 ∪ ran ∪ ran (comp‘𝑌) ∈ V
88	81, 87	xpex 7765	. . . . . . . . . . 11 ⊢ (𝒫 ∪ ran ∪ ran (comp‘𝑋) × 𝒫 ∪ ran ∪ ran (comp‘𝑌)) ∈ V
89		fvex 6896	. . . . . . . . . . . . . 14 ⊢ (Hom ‘𝑇) ∈ V
90	89	rnex 7920	. . . . . . . . . . . . 13 ⊢ ran (Hom ‘𝑇) ∈ V
91	90	uniex 7756	. . . . . . . . . . . 12 ⊢ ∪ ran (Hom ‘𝑇) ∈ V
92	91, 91	xpex 7765	. . . . . . . . . . 11 ⊢ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇)) ∈ V
93		ovssunirn 7454	. . . . . . . . . . . . . . . 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))
94		ovssunirn 7454	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦)) ⊆ ∪ ran (comp‘𝑋)
95		rnss 5921	. . . . . . . . . . . . . . . . 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))
96		uniss 4875	. . . . . . . . . . . . . . . . 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))
97	94, 95, 96	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran (⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦)) ⊆ ∪ ran ∪ ran (comp‘𝑋)
98	93, 97	sstri 3940	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑋)
99		ovex 7451	. . . . . . . . . . . . . . . 16 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ V
100	99	elpw 4561	. . . . . . . . . . . . . . 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))
101	98, 100	mpbir 234	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑔)(⟨(1st ‘(1st ‘𝑥)), (1st ‘(2nd ‘𝑥))⟩(comp‘𝑋)(1st ‘𝑦))(1st ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑋)
102		ovssunirn 7454	. . . . . . . . . . . . . . . 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))
103		ovssunirn 7454	. . . . . . . . . . . . . . . . 17 ⊢ (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦)) ⊆ ∪ ran (comp‘𝑌)
104		rnss 5921	. . . . . . . . . . . . . . . . 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))
105		uniss 4875	. . . . . . . . . . . . . . . . 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))
106	103, 104, 105	mp2b 10	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran (⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
107	102, 106	sstri 3940	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ⊆ ∪ ran ∪ ran (comp‘𝑌)
108		ovex 7451	. . . . . . . . . . . . . . . 16 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ V
109	108	elpw 4561	. . . . . . . . . . . . . . 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))
110	107, 109	mpbir 234	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑔)(⟨(2nd ‘(1st ‘𝑥)), (2nd ‘(2nd ‘𝑥))⟩(comp‘𝑌)(2nd ‘𝑦))(2nd ‘𝑓)) ∈ 𝒫 ∪ ran ∪ ran (comp‘𝑌)
111		opelxpi 5688	. . . . . . . . . . . . . 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)))
112	101, 110, 111	mp2an 705	. . . . . . . . . . . . 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))
113	112	rgen2w 3082	. . . . . . . . . . . 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))
114		eqid 2761	. . . . . . . . . . . . 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))⟩)
115	114	fmpo 8077	. . . . . . . . . . . 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)))
116	113, 115	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))
117		ovssunirn 7454	. . . . . . . . . . . 12 ⊢ ((2nd ‘𝑥)(Hom ‘𝑇)𝑦) ⊆ ∪ ran (Hom ‘𝑇)
118		fvssunirn 6914	. . . . . . . . . . . 12 ⊢ ((Hom ‘𝑇)‘𝑥) ⊆ ∪ ran (Hom ‘𝑇)
119		xpss12 5666	. . . . . . . . . . . 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)))
120	117, 118, 119	mp2an 705	. . . . . . . . . . 11 ⊢ (((2nd ‘𝑥)(Hom ‘𝑇)𝑦) × ((Hom ‘𝑇)‘𝑥)) ⊆ (∪ ran (Hom ‘𝑇) × ∪ ran (Hom ‘𝑇))
121		elpm2r 8858	. . . . . . . . . . 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))))
122	88, 92, 116, 120, 121	mp4an 706	. . . . . . . . . 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)))
123	122	rgen2w 3082	. . . . . . . . 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)))
124		eqid 2761	. . . . . . . . . 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))⟩))
125	124	fmpo 8077	. . . . . . . . 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))))
126	123, 125	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)))
127	126	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))))
128	17, 58, 75, 127	wunf 10805	. . . . . 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)
129	17, 57, 128	wunop 10800	. . . . 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)
130	17, 27, 55, 129	wuntp 10789	. . . 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)
131	16, 130	eqeltrd 2861	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑈)
132	22, 23, 17	catcbas 18269	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
133	8, 132	eleqtrd 2863	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat))
134	133	elin2d 4151	. . . 4 ⊢ (𝜑 → 𝑋 ∈ Cat)
135	9, 132	eleqtrd 2863	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat))
136	135	elin2d 4151	. . . 4 ⊢ (𝜑 → 𝑌 ∈ Cat)
137	1, 134, 136	xpccat 18357	. . 3 ⊢ (𝜑 → 𝑇 ∈ Cat)
138	131, 137	elind 4146	. 2 ⊢ (𝜑 → 𝑇 ∈ (𝑈 ∩ Cat))
139	138, 132	eleqtrrd 2864	1 ⊢ (𝜑 → 𝑇 ∈ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  {ctp 4588  ⟨cop 4590  ∪ cuni 4867   × cxp 5649  ran crn 5652  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  ωcom 7875  1^st c1st 7997  2^nd c2nd 7998   ↑_pm cpm 8841  WUnicwun 10778  ndxcnx 17364  Basecbs 17380  Hom chom 17432  compcco 17433  Catccat 17831  CatCatccatc 18266   ×_c cxpc 18335

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-oadd 8473  df-omul 8474  df-er 8710  df-ec 8712  df-qs 8716  df-map 8842  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-wun 10780  df-ni 10950  df-pli 10951  df-mi 10952  df-lti 10953  df-plpq 10986  df-mpq 10987  df-ltpq 10988  df-enq 10989  df-nq 10990  df-erq 10991  df-plq 10992  df-mq 10993  df-1nq 10994  df-rq 10995  df-ltnq 10996  df-np 11059  df-plp 11061  df-ltp 11063  df-enr 11133  df-nr 11134  df-c 11199  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-z 12687  df-dec 12808  df-uz 12959  df-fz 13633  df-struct 17318  df-slot 17353  df-ndx 17365  df-base 17381  df-hom 17445  df-cco 17446  df-cat 17835  df-cid 17836  df-catc 18267  df-xpc 18339

This theorem is used by: (None)