Theorem funcringcsetc 20661

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 2729	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2		funcringcsetc.s	. . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈)
3		eqid 2729	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4		eqid 2729	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		funcringcsetc.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	1, 5	estrcbas 18156	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
7	6	mpteq1d 5248	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8		mpoeq12 7499	. . . . . . 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 582	. . . . . 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 18181	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
11		df-br 5154	. . . . 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		funcringcsetc.r	. . . . . . 7 ⊢ 𝑅 = (RingCat‘𝑈)
14		eqid 2729	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14, 5	ringcbas 20637	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (𝑈 ∩ Ring))
16		incom 4209	. . . . . 6 ⊢ (𝑈 ∩ Ring) = (Ring ∩ 𝑈)
17	15, 16	eqtrdi 2785	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Ring ∩ 𝑈))
18		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
19	13, 14, 5, 18	ringchomfval 20638	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) = ( RingHom ↾ ((Base‘𝑅) × (Base‘𝑅))))
20	1, 5, 17, 19	rhmsubcsetc 20649	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
21	12, 20	funcres 17923	. . 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 7239	. . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
23	5, 22	syl 17	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
24		fvex 6915	. . . . . 6 ⊢ (Hom ‘𝑅) ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) ∈ V)
26		mpoexga 8094	. . . . . 6 ⊢ ((𝑈 ∈ WUni ∧ 𝑈 ∈ WUni) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
27	5, 5, 26	syl2anc 582	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
28	15, 19	rhmresfn 20635	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
29	23, 25, 27, 28	resfval2 17920	. . . 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 4237	. . . . . . . 8 ⊢ (𝑈 ∩ Ring) ⊆ 𝑈
31	15, 30	eqsstrdi 4043	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ⊆ 𝑈)
32	31	resmptd 6051	. . . . . 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 5248	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
37	33, 36	eqtr2d 2770	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
38	32, 37	eqtrd 2769	. . . . 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 7432	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 RingHom 𝑦) = (𝑎 RingHom 𝑦))
41	40	reseq2d 5991	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ( I ↾ (𝑥 RingHom 𝑦)) = ( I ↾ (𝑎 RingHom 𝑦)))
42		oveq2 7433	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑎 RingHom 𝑦) = (𝑎 RingHom 𝑏))
43	42	reseq2d 5991	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ( I ↾ (𝑎 RingHom 𝑦)) = ( I ↾ (𝑎 RingHom 𝑏)))
44	41, 43	cbvmpov 7521	. . . . . . 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 2730	. . . . . . . . . 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 6902	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49		fveq2 6902	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
50	48, 49	oveqan12rd 7445	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
51	50	reseq2d 5991	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
52	51	adantl 480	. . . . . . . . . 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 4037	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
54	53	sseld 3987	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → 𝑎 ∈ 𝑈))
55	54	com12 32	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐵 → (𝜑 → 𝑎 ∈ 𝑈))
56	55	adantr 479	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑎 ∈ 𝑈))
57	56	impcom 406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝑈)
58	53	sseld 3987	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑈))
59	58	adantld 489	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑈))
60	59	imp 405	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝑈)
61		ovexd 7460	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
62	61	resiexd 7234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ∈ V)
63	47, 52, 57, 60, 62	ovmpod 7578	. . . . . . . . 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 5990	. . . . . . . 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 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑈 ∈ WUni)
66		simprl 769	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
67		simprr 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
68	13, 34, 65, 18, 66, 67	ringchom 20639	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RingHom 𝑏))
69	68	reseq2d 5991	. . . . . . . 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 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝑎) = (Base‘𝑎)
71		eqid 2729	. . . . . . . . . . . 12 ⊢ (Base‘𝑏) = (Base‘𝑏)
72	70, 71	rhmf 20476	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑎 RingHom 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
73		fvex 6915	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑏) ∈ V
74		fvex 6915	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑎) ∈ V
75	73, 74	pm3.2i 469	. . . . . . . . . . . . 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 8871	. . . . . . . . . . . 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 RingHom 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
80	79	ssrdv 3994	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
81	80	resabs1d 6019	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RingHom 𝑏)) = ( I ↾ (𝑎 RingHom 𝑏)))
82	64, 69, 81	3eqtrrd 2774	. . . . . . 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 7500	. . . . . 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 2774	. . . . 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 4889	. . . 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 2770	. . 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 20634	. . . 4 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
88	87	oveq1d 7440	. . 3 ⊢ (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
89	21, 86, 88	3eltr4d 2844	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90		df-br 5154	. 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 394   = wceq 1534   ∈ wcel 2100  Vcvv 3471   ∩ cin 3955  ⟨cop 4639   class class class wbr 5153   ↦ cmpt 5236   I cid 5580   ↾ cres 5685  ⟶wf 6551  ‘cfv 6555  (class class class)co 7425   ∈ cmpo 7427   ↑_m cmap 8858  WUnicwun 10747  Basecbs 17221  Hom chom 17285   ↾_cat cresc 17832   Func cfunc 17881   ↾_f cresf 17884  SetCatcsetc 18105  ExtStrCatcestrc 18153  Ringcrg 20225   RingHom crh 20460  RingCatcringc 20632

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 2102  ax-9 2110  ax-10 2133  ax-11 2150  ax-12 2170  ax-ext 2700  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5370  ax-pr 5434  ax-un 7746  ax-cnex 11218  ax-resscn 11219  ax-1cn 11220  ax-icn 11221  ax-addcl 11222  ax-addrcl 11223  ax-mulcl 11224  ax-mulrcl 11225  ax-mulcom 11226  ax-addass 11227  ax-mulass 11228  ax-distr 11229  ax-i2m1 11230  ax-1ne0 11231  ax-1rid 11232  ax-rnegex 11233  ax-rrecex 11234  ax-cnre 11235  ax-pre-lttri 11236  ax-pre-lttrn 11237  ax-pre-ltadd 11238  ax-pre-mulgt0 11239

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2062  df-mo 2532  df-eu 2561  df-clab 2707  df-cleq 2721  df-clel 2806  df-nfc 2881  df-ne 2934  df-nel 3040  df-ral 3055  df-rex 3064  df-rmo 3372  df-reu 3373  df-rab 3428  df-v 3473  df-sbc 3786  df-csb 3902  df-dif 3959  df-un 3961  df-in 3963  df-ss 3973  df-pss 3976  df-nul 4333  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-tp 4638  df-op 4640  df-uni 4916  df-iun 5005  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5581  df-eprel 5587  df-po 5595  df-so 5596  df-fr 5638  df-we 5640  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6314  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6507  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7381  df-ov 7428  df-oprab 7429  df-mpo 7430  df-om 7880  df-1st 8006  df-2nd 8007  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-1o 8499  df-er 8737  df-map 8860  df-pm 8861  df-ixp 8930  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-wun 10749  df-pnf 11304  df-mnf 11305  df-xr 11306  df-ltxr 11307  df-le 11308  df-sub 11500  df-neg 11501  df-nn 12272  df-2 12334  df-3 12335  df-4 12336  df-5 12337  df-6 12338  df-7 12339  df-8 12340  df-9 12341  df-n0 12532  df-z 12618  df-dec 12737  df-uz 12882  df-fz 13546  df-struct 17157  df-sets 17174  df-slot 17192  df-ndx 17204  df-base 17222  df-ress 17251  df-plusg 17287  df-hom 17298  df-cco 17299  df-0g 17464  df-cat 17689  df-cid 17690  df-homf 17691  df-ssc 17834  df-resc 17835  df-subc 17836  df-func 17885  df-resf 17888  df-setc 18106  df-estrc 18154  df-mgm 18641  df-sgrp 18720  df-mnd 18736  df-mhm 18781  df-grp 18939  df-ghm 19216  df-mgp 20127  df-ur 20174  df-ring 20227  df-rhm 20463  df-ringc 20633

This theorem is referenced by: (None)