Theorem funcringcsetc 20691

Description: The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 26-Mar-2020.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem funcringcsetc

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

Step	Hyp	Ref	Expression
1		eqid 2734	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2		funcringcsetc.s	. . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈)
3		eqid 2734	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4		eqid 2734	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		funcringcsetc.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	1, 5	estrcbas 18188	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
7	6	mpteq1d 5264	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8		mpoeq12 7519	. . . . . . 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 18213	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
11		df-br 5170	. . . . 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		funcringcsetc.r	. . . . . . 7 ⊢ 𝑅 = (RingCat‘𝑈)
14		eqid 2734	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14, 5	ringcbas 20667	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (𝑈 ∩ Ring))
16		incom 4224	. . . . . 6 ⊢ (𝑈 ∩ Ring) = (Ring ∩ 𝑈)
17	15, 16	eqtrdi 2790	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Ring ∩ 𝑈))
18		eqid 2734	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
19	13, 14, 5, 18	ringchomfval 20668	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) = ( RingHom ↾ ((Base‘𝑅) × (Base‘𝑅))))
20	1, 5, 17, 19	rhmsubcsetc 20679	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
21	12, 20	funcres 17955	. . 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 7256	. . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
23	5, 22	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
24		fvex 6932	. . . . . 6 ⊢ (Hom ‘𝑅) ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) ∈ V)
26		mpoexga 8114	. . . . . 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	rhmresfn 20665	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
29	23, 25, 27, 28	resfval2 17952	. . . 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 4252	. . . . . . . 8 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈
31	15, 30	eqsstrdi 4057	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ⊆ 𝑈)
32	31	resmptd 6068	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
33		funcringcsetc.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
34		funcringcsetc.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅)
35	34	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
36	35	mpteq1d 5264	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
37	33, 36	eqtr2d 2775	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
38	32, 37	eqtrd 2774	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ↾ (Base‘𝑅)) = 𝐹)
39		funcringcsetc.g	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))
40		oveq1 7452	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 RingHom 𝑦) = (𝑎 RingHom 𝑦))
41	40	reseq2d 6008	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ( I ↾ (𝑥 RingHom 𝑦)) = ( I ↾ (𝑎 RingHom 𝑦)))
42		oveq2 7453	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑎 RingHom 𝑦) = (𝑎 RingHom 𝑏))
43	42	reseq2d 6008	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ( I ↾ (𝑎 RingHom 𝑦)) = ( I ↾ (𝑎 RingHom 𝑏)))
44	41, 43	cbvmpov 7541	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ ( I ↾ (𝑎 RingHom 𝑏)))
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))) = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ ( I ↾ (𝑎 RingHom 𝑏))))
46	34	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝐵 = (Base‘𝑅))
47		eqidd 2735	. . . . . . . . . 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 6919	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49		fveq2 6919	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
50	48, 49	oveqan12rd 7465	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
51	50	reseq2d 6008	. . . . . . . . . . 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 4051	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
54	53	sseld 4001	. . . . . . . . . . . . 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 4001	. . . . . . . . . . . 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 7480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
62	61	resiexd 7251	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ∈ V)
63	47, 52, 57, 60, 62	ovmpod 7598	. . . . . . . . 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 6007	. . . . . . . 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	ringchom 20669	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RingHom 𝑏))
69	68	reseq2d 6008	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎(Hom ‘𝑅)𝑏)) = (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RingHom 𝑏)))
70		eqid 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝑎) = (Base‘𝑎)
71		eqid 2734	. . . . . . . . . . . 12 ⊢ (Base‘𝑏) = (Base‘𝑏)
72	70, 71	rhmf 20506	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑎 RingHom 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
73		fvex 6932	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑏) ∈ V
74		fvex 6932	. . . . . . . . . . . . . 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 8893	. . . . . . . . . . . 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 RingHom 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
80	79	ssrdv 4008	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
81	80	resabs1d 6036	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RingHom 𝑏)) = ( I ↾ (𝑎 RingHom 𝑏)))
82	64, 69, 81	3eqtrrd 2779	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( I ↾ (𝑎 RingHom 𝑏)) = ((𝑎(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))𝑏) ↾ (𝑎(Hom ‘𝑅)𝑏)))
83	35, 46, 82	mpoeq123dva 7520	. . . . . 6 ⊢ (𝜑 → (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ ( I ↾ (𝑎 RingHom 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 2779	. . . . 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 4905	. . . 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 2775	. . 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	ringcval 20664	. . . 4 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
88	87	oveq1d 7460	. . 3 ⊢ (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
89	21, 86, 88	3eltr4d 2853	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90		df-br 5170	. 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 1537   ∈ wcel 2103  Vcvv 3482   ∩ cin 3969  ⟨cop 4654   class class class wbr 5169   ↦ cmpt 5252   I cid 5596   ↾ cres 5701  ⟶wf 6568  ‘cfv 6572  (class class class)co 7445   ∈ cmpo 7447   ↑_m cmap 8880  WUnicwun 10765  Basecbs 17253  Hom chom 17317   ↾_cat cresc 17864   Func cfunc 17913   ↾_f cresf 17916  SetCatcsetc 18137  ExtStrCatcestrc 18185  Ringcrg 20255   RingHom crh 20490  RingCatcringc 20662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-rmo 3383  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-er 8759  df-map 8882  df-pm 8883  df-ixp 8952  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-wun 10767  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-nn 12290  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12550  df-z 12636  df-dec 12755  df-uz 12900  df-fz 13564  df-struct 17189  df-sets 17206  df-slot 17224  df-ndx 17236  df-base 17254  df-ress 17283  df-plusg 17319  df-hom 17330  df-cco 17331  df-0g 17496  df-cat 17721  df-cid 17722  df-homf 17723  df-ssc 17866  df-resc 17867  df-subc 17868  df-func 17917  df-resf 17920  df-setc 18138  df-estrc 18186  df-mgm 18673  df-sgrp 18752  df-mnd 18768  df-mhm 18813  df-grp 18971  df-ghm 19248  df-mgp 20157  df-ur 20204  df-ring 20257  df-rhm 20493  df-ringc 20663

This theorem is referenced by: (None)