Theorem funcrngcsetc 20608

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 20609, using cofuval2 17845 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 20607, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18106. (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 2737	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2		funcrngcsetc.s	. . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈)
3		eqid 2737	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4		eqid 2737	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		funcrngcsetc.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	1, 5	estrcbas 18082	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
7	6	mpteq1d 5176	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8		mpoeq12 7433	. . . . . . 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 585	. . . . . 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 18106	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
11		df-br 5087	. . . . 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 218	. . . 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 2737	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14, 5	rngcbas 20589	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (𝑈 ∩ Rng))
16		incom 4150	. . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
17	15, 16	eqtrdi 2788	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Rng ∩ 𝑈))
18		eqid 2737	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
19	13, 14, 5, 18	rngchomfval 20590	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ ((Base‘𝑅) × (Base‘𝑅))))
20	1, 5, 17, 19	rnghmsubcsetc 20601	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
21	12, 20	funcres 17854	. . 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 7169	. . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
23	5, 22	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
24		fvex 6847	. . . . . 6 ⊢ (Hom ‘𝑅) ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) ∈ V)
26		mpoexga 8023	. . . . . 6 ⊢ ((𝑈 ∈ WUni ∧ 𝑈 ∈ WUni) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
27	5, 5, 26	syl2anc 585	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
28	15, 19	rnghmresfn 20587	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
29	23, 25, 27, 28	resfval2 17851	. . . 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 4178	. . . . . . . 8 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
31	15, 30	eqsstrdi 3967	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ⊆ 𝑈)
32	31	resmptd 5999	. . . . . 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 5176	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
37	33, 36	eqtr2d 2773	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
38	32, 37	eqtrd 2772	. . . . 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 7367	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 RngHom 𝑦) = (𝑎 RngHom 𝑦))
41	40	reseq2d 5938	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ( I ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑎 RngHom 𝑦)))
42		oveq2 7368	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑎 RngHom 𝑦) = (𝑎 RngHom 𝑏))
43	42	reseq2d 5938	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ( I ↾ (𝑎 RngHom 𝑦)) = ( I ↾ (𝑎 RngHom 𝑏)))
44	41, 43	cbvmpov 7455	. . . . . . 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 2738	. . . . . . . . . 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 6834	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49		fveq2 6834	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
50	48, 49	oveqan12rd 7380	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
51	50	reseq2d 5938	. . . . . . . . . . 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 3961	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
54	53	sseld 3921	. . . . . . . . . . . . 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 3921	. . . . . . . . . . . 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 7395	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
62	61	resiexd 7164	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ∈ V)
63	47, 52, 57, 60, 62	ovmpod 7512	. . . . . . . . 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 5937	. . . . . . . 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 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
67		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
68	13, 34, 65, 18, 66, 67	rngchom 20591	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RngHom 𝑏))
69	68	reseq2d 5938	. . . . . . . 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 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝑎) = (Base‘𝑎)
71		eqid 2737	. . . . . . . . . . . 12 ⊢ (Base‘𝑏) = (Base‘𝑏)
72	70, 71	rnghmf 20419	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑎 RngHom 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
73		fvex 6847	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑏) ∈ V
74		fvex 6847	. . . . . . . . . . . . . 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 8779	. . . . . . . . . . . 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 246	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑓 ∈ (𝑎 RngHom 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
80	79	ssrdv 3928	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
81	80	resabs1d 5967	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RngHom 𝑏)) = ( I ↾ (𝑎 RngHom 𝑏)))
82	64, 69, 81	3eqtrrd 2777	. . . . . . 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 7434	. . . . . 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 2777	. . . . 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 4825	. . . 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 2773	. . 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 20586	. . . 4 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
88	87	oveq1d 7375	. . 3 ⊢ (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
89	21, 86, 88	3eltr4d 2852	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90		df-br 5087	. 2 ⊢ (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
91	89, 90	sylibr 234	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ∩ cin 3889  ⟨cop 4574   class class class wbr 5086   ↦ cmpt 5167   I cid 5518   ↾ cres 5626  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360   ∈ cmpo 7362   ↑_m cmap 8766  WUnicwun 10614  Basecbs 17170  Hom chom 17222   ↾_cat cresc 17766   Func cfunc 17812   ↾_f cresf 17815  SetCatcsetc 18033  ExtStrCatcestrc 18079  Rngcrng 20124   RngHom crnghm 20405  RngCatcrngc 20584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-er 8636  df-map 8768  df-pm 8769  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-wun 10616  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-hom 17235  df-cco 17236  df-0g 17395  df-cat 17625  df-cid 17626  df-homf 17627  df-ssc 17768  df-resc 17769  df-subc 17770  df-func 17816  df-resf 17819  df-setc 18034  df-estrc 18080  df-mgm 18599  df-mgmhm 18651  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-grp 18903  df-ghm 19179  df-abl 19749  df-mgp 20113  df-rng 20125  df-rnghm 20407  df-rngc 20585

This theorem is referenced by: (None)