Theorem funcrngcsetc 20555

Description: The "natural forgetful functor" from the category of non-unital rings into the category of sets which sends each non-unital ring to its underlying set (base set) and the morphisms (non-unital ring homomorphisms) to mappings of the corresponding base sets. An alternate proof is provided in funcrngcsetcALT 20556, using cofuval2 17858 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 20554, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18125. (Contributed by AV, 26-Mar-2020.)

Hypotheses

Ref	Expression
funcrngcsetc.r	⊢ 𝑅 = (RngCat‘𝑈)
funcrngcsetc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcrngcsetc.b	⊢ 𝐵 = (Base‘𝑅)
funcrngcsetc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcrngcsetc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
funcrngcsetc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))))

Assertion

Ref	Expression
funcrngcsetc	⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆   𝑥,𝑈,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑆(𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem funcrngcsetc

Dummy variables 𝑎 𝑏 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2727	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2		funcrngcsetc.s	. . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈)
3		eqid 2727	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4		eqid 2727	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		funcrngcsetc.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	1, 5	estrcbas 18100	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
7	6	mpteq1d 5237	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8		mpoeq12 7487	. . . . . . 7 ⊢ ((𝑈 = (Base‘(ExtStrCat‘𝑈)) ∧ 𝑈 = (Base‘(ExtStrCat‘𝑈))) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑦 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
9	6, 6, 8	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑦 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
10	1, 2, 3, 4, 5, 7, 9	funcestrcsetc 18125	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
11		df-br 5143	. . . . 5 ⊢ ((𝑥 ∈ 𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ↔ ⟨(𝑥 ∈ 𝑈 ↦ (Base‘𝑥)), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
12	10, 11	sylib 217	. . . 4 ⊢ (𝜑 → ⟨(𝑥 ∈ 𝑈 ↦ (Base‘𝑥)), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
13		funcrngcsetc.r	. . . . . . 7 ⊢ 𝑅 = (RngCat‘𝑈)
14		eqid 2727	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14, 5	rngcbas 20536	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (𝑈 ∩ Rng))
16		incom 4197	. . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
17	15, 16	eqtrdi 2783	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Rng ∩ 𝑈))
18		eqid 2727	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
19	13, 14, 5, 18	rngchomfval 20537	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ ((Base‘𝑅) × (Base‘𝑅))))
20	1, 5, 17, 19	rnghmsubcsetc 20548	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
21	12, 20	funcres 17867	. . 3 ⊢ (𝜑 → (⟨(𝑥 ∈ 𝑈 ↦ (Base‘𝑥)), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)) ∈ (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
22		mptexg 7227	. . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
23	5, 22	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
24		fvex 6904	. . . . . 6 ⊢ (Hom ‘𝑅) ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) ∈ V)
26		mpoexga 8074	. . . . . 6 ⊢ ((𝑈 ∈ WUni ∧ 𝑈 ∈ WUni) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
27	5, 5, 26	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
28	15, 19	rnghmresfn 20534	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
29	23, 25, 27, 28	resfval2 17864	. . . 4 ⊢ (𝜑 → (⟨(𝑥 ∈ 𝑈 ↦ (Base‘𝑥)), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)) = ⟨((𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)), (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))⟩)
30		inss1 4224	. . . . . . . 8 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
31	15, 30	eqsstrdi 4032	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ⊆ 𝑈)
32	31	resmptd 6038	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
33		funcrngcsetc.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
34		funcrngcsetc.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
35	34	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
36	35	mpteq1d 5237	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
37	33, 36	eqtr2d 2768	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
38	32, 37	eqtrd 2767	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = 𝐹)
39		funcrngcsetc.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))))
40		oveq1 7421	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 RngHom 𝑦) = (𝑎 RngHom 𝑦))
41	40	reseq2d 5979	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ( I ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑎 RngHom 𝑦)))
42		oveq2 7422	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑎 RngHom 𝑦) = (𝑎 RngHom 𝑏))
43	42	reseq2d 5979	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ( I ↾ (𝑎 RngHom 𝑦)) = ( I ↾ (𝑎 RngHom 𝑏)))
44	41, 43	cbvmpov 7509	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))) = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ ( I ↾ (𝑎 RngHom 𝑏)))
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))) = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ ( I ↾ (𝑎 RngHom 𝑏))))
46	34	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝑅))
47		eqidd 2728	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
48		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
50	48, 49	oveqan12rd 7434	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
51	50	reseq2d 5979	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
52	51	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) ∧ (𝑥 = 𝑎 ∧ 𝑦 = 𝑏)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
53	34, 31	eqsstrid 4026	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
54	53	sseld 3977	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → 𝑎 ∈ 𝑈))
55	54	com12 32	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐵 → (𝜑 → 𝑎 ∈ 𝑈))
56	55	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑎 ∈ 𝑈))
57	56	impcom 407	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝑈)
58	53	sseld 3977	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑈))
59	58	adantld 490	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑈))
60	59	imp 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝑈)
61		ovexd 7449	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
62	61	resiexd 7222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ∈ V)
63	47, 52, 57, 60, 62	ovmpod 7565	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
64	63	reseq1d 5978	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)))
65	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑈 ∈ WUni)
66		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
67		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
68	13, 34, 65, 18, 66, 67	rngchom 20538	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RngHom 𝑏))
69	68	reseq2d 5979	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RngHom 𝑏)))
70		eqid 2727	. . . . . . . . . . . 12 ⊢ (Base‘𝑎) = (Base‘𝑎)
71		eqid 2727	. . . . . . . . . . . 12 ⊢ (Base‘𝑏) = (Base‘𝑏)
72	70, 71	rnghmf 20369	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑎 RngHom 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
73		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑏) ∈ V
74		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑎) ∈ V
75	73, 74	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
76	75	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V))
77		elmapg 8847	. . . . . . . . . . . 12 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
78	76, 77	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
79	72, 78	imbitrrid 245	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑓 ∈ (𝑎 RngHom 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
80	79	ssrdv 3984	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
81	80	resabs1d 6010	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RngHom 𝑏)) = ( I ↾ (𝑎 RngHom 𝑏)))
82	64, 69, 81	3eqtrrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( I ↾ (𝑎 RngHom 𝑏)) = ((𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))
83	35, 46, 82	mpoeq123dva 7488	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ ( I ↾ (𝑎 RngHom 𝑏))) = (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))))
84	39, 45, 83	3eqtrrd 2772	. . . . 5 ⊢ (𝜑 → (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏))) = 𝐺)
85	38, 84	opeq12d 4877	. . . 4 ⊢ (𝜑 → ⟨((𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)), (𝑎 ∈ (Base‘𝑅), 𝑏 ∈ (Base‘𝑅) ↦ ((𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))⟩ = ⟨𝐹, 𝐺⟩)
86	29, 85	eqtr2d 2768	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = (⟨(𝑥 ∈ 𝑈 ↦ (Base‘𝑥)), (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))⟩ ↾f (Hom ‘𝑅)))
87	13, 5, 15, 19	rngcval 20533	. . . 4 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
88	87	oveq1d 7429	. . 3 ⊢ (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
89	21, 86, 88	3eltr4d 2843	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90		df-br 5143	. 2 ⊢ (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
91	89, 90	sylibr 233	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ∩ cin 3943  ⟨cop 4630   class class class wbr 5142   ↦ cmpt 5225   I cid 5569   ↾ cres 5674  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416   ↑_m cmap 8834  WUnicwun 10709  Basecbs 17165  Hom chom 17229   ↾_cat cresc 17776   Func cfunc 17825   ↾_f cresf 17828  SetCatcsetc 18049  ExtStrCatcestrc 18097  Rngcrng 20076   RngHom crnghm 20355  RngCatcrngc 20531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-pm 8837  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-wun 10711  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-fz 13503  df-struct 17101  df-sets 17118  df-slot 17136  df-ndx 17148  df-base 17166  df-ress 17195  df-plusg 17231  df-hom 17242  df-cco 17243  df-0g 17408  df-cat 17633  df-cid 17634  df-homf 17635  df-ssc 17778  df-resc 17779  df-subc 17780  df-func 17829  df-resf 17832  df-setc 18050  df-estrc 18098  df-mgm 18585  df-mgmhm 18637  df-sgrp 18664  df-mnd 18680  df-mhm 18725  df-grp 18878  df-ghm 19152  df-abl 19722  df-mgp 20059  df-rng 20077  df-rnghm 20357  df-rngc 20532

This theorem is referenced by: (None)