Theorem funcrngcsetc 20898

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 20899, using cofuval2 18068 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 20897, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18329. (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 2760	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
2		funcrngcsetc.s	. . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈)
3		eqid 2760	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
4		eqid 2760	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
5		funcrngcsetc.u	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	1, 5	estrcbas 18305	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
7	6	mpteq1d 5194	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑥)))
8		mpoeq12 7494	. . . . . . 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 596	. . . . . 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 18329	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥))((ExtStrCat‘𝑈) Func 𝑆)(𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))))
11		df-br 5103	. . . . 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 221	. . . 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 2760	. . . . . . 7 ⊢ (Base‘𝑅) = (Base‘𝑅)
15	13, 14, 5	rngcbas 20879	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (𝑈 ∩ Rng))
16		incom 4106	. . . . . 6 ⊢ (𝑈 ∩ Rng) = (Rng ∩ 𝑈)
17	15, 16	eqtrdi 2811	. . . . 5 ⊢ (𝜑 → (Base‘𝑅) = (Rng ∩ 𝑈))
18		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝑅) = (Hom ‘𝑅)
19	13, 14, 5, 18	rngchomfval 20880	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) = ( RngHom ↾ ((Base‘𝑅) × (Base‘𝑅))))
20	1, 5, 17, 19	rnghmsubcsetc 20891	. . . 4 ⊢ (𝜑 → (Hom ‘𝑅) ∈ (Subcat‘(ExtStrCat‘𝑈)))
21	12, 20	funcres 18077	. . 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 7228	. . . . . 6 ⊢ (𝑈 ∈ WUni → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
23	5, 22	syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈 ↦ (Base‘𝑥)) ∈ V)
24		fvex 6899	. . . . . 6 ⊢ (Hom ‘𝑅) ∈ V
25	24	a1i 11	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) ∈ V)
26		mpoexga 8091	. . . . . 6 ⊢ ((𝑈 ∈ WUni ∧ 𝑈 ∈ WUni) → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
27	5, 5, 26	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥)))) ∈ V)
28	15, 19	rnghmresfn 20877	. . . . 5 ⊢ (𝜑 → (Hom ‘𝑅) Fn ((Base‘𝑅) × (Base‘𝑅)))
29	23, 25, 27, 28	resfval2 18074	. . . 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 4133	. . . . . . . 8 ⊢ (𝑈 ∩ Rng) ⊆ 𝑈
31	15, 30	eqsstrdi 3952	. . . . . . 7 ⊢ (𝜑 → (Base‘𝑅) ⊆ 𝑈)
32	31	resmptd 6031	. . . . . 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 5194	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)))
37	33, 36	eqtr2d 2796	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑅) ↦ (Base‘𝑥)) = 𝐹)
38	32, 37	eqtrd 2795	. . . . 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 7428	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → (𝑥 RngHom 𝑦) = (𝑎 RngHom 𝑦))
41	40	reseq2d 5969	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → ( I ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑎 RngHom 𝑦)))
42		oveq2 7429	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (𝑎 RngHom 𝑦) = (𝑎 RngHom 𝑏))
43	42	reseq2d 5969	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → ( I ↾ (𝑎 RngHom 𝑦)) = ( I ↾ (𝑎 RngHom 𝑏)))
44	41, 43	cbvmpov 7516	. . . . . . 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 2761	. . . . . . . . . 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 6886	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
49		fveq2 6886	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
50	48, 49	oveqan12rd 7441	. . . . . . . . . . . 12 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑏) ↑m (Base‘𝑎)))
51	50	reseq2d 5969	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))))
52	51	adantl 487	. . . . . . . . . 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 3946	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
54	53	sseld 3907	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 ∈ 𝐵 → 𝑎 ∈ 𝑈))
55	54	com12 33	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ 𝐵 → (𝜑 → 𝑎 ∈ 𝑈))
56	55	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝜑 → 𝑎 ∈ 𝑈))
57	56	impcom 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝑈)
58	53	sseld 3907	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ 𝑈))
59	58	adantld 496	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑈))
60	59	imp 412	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝑈)
61		ovexd 7456	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((Base‘𝑏) ↑m (Base‘𝑎)) ∈ V)
62	61	resiexd 7223	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → ( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ∈ V)
63	47, 52, 57, 60, 62	ovmpod 7573	. . . . . . . . 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 5968	. . . . . . . 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 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑈 ∈ WUni)
66		simprl 783	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑎 ∈ 𝐵)
67		simprr 785	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → 𝑏 ∈ 𝐵)
68	13, 34, 65, 18, 66, 67	rngchom 20881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎(Hom ‘𝑅)𝑏) = (𝑎 RngHom 𝑏))
69	68	reseq2d 5969	. . . . . . . 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 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝑎) = (Base‘𝑎)
71		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝑏) = (Base‘𝑏)
72	70, 71	rnghmf 20684	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (𝑎 RngHom 𝑏) → 𝑓:(Base‘𝑎)⟶(Base‘𝑏))
73		fvex 6899	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑏) ∈ V
74		fvex 6899	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑎) ∈ V
75	73, 74	pm3.2i 476	. . . . . . . . . . . . 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 8863	. . . . . . . . . . . 12 ⊢ (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → (𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
78	76, 77	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎)) ↔ 𝑓:(Base‘𝑎)⟶(Base‘𝑏)))
79	72, 78	imbitrrid 249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑓 ∈ (𝑎 RngHom 𝑏) → 𝑓 ∈ ((Base‘𝑏) ↑m (Base‘𝑎))))
80	79	ssrdv 3914	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (𝑎 RngHom 𝑏) ⊆ ((Base‘𝑏) ↑m (Base‘𝑎)))
81	80	resabs1d 5998	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵)) → (( I ↾ ((Base‘𝑏) ↑m (Base‘𝑎))) ↾ (𝑎 RngHom 𝑏)) = ( I ↾ (𝑎 RngHom 𝑏)))
82	64, 69, 81	3eqtrrd 2800	. . . . . . 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 7495	. . . . . 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 2800	. . . . 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 4847	. . . 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 2796	. . 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 20876	. . . 4 ⊢ (𝜑 → 𝑅 = ((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)))
88	87	oveq1d 7436	. . 3 ⊢ (𝜑 → (𝑅 Func 𝑆) = (((ExtStrCat‘𝑈) ↾cat (Hom ‘𝑅)) Func 𝑆))
89	21, 86, 88	3eltr4d 2875	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
90		df-br 5103	. 2 ⊢ (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
91	89, 90	sylibr 237	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450   ∩ cin 4089  ⟨cop 4834   class class class wbr 5102   ↦ cmpt 5185   I cid 5544   ↾ cres 5652  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423   ↑_m cmap 8851  WUnicwun 10789  Basecbs 17393  Hom chom 17445   ↾_cat cresc 17989   Func cfunc 18035   ↾_f cresf 18038  SetCatcsetc 18256  ExtStrCatcestrc 18302  Rngcrng 20380   RngHom crnghm 20670  RngCatcrngc 20874

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-er 8721  df-map 8853  df-pm 8854  df-ixp 8930  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-wun 10791  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-2 12409  df-3 12410  df-4 12411  df-5 12412  df-6 12413  df-7 12414  df-8 12415  df-9 12416  df-n0 12611  df-z 12698  df-dec 12819  df-uz 12970  df-fz 13644  df-struct 17331  df-sets 17348  df-slot 17366  df-ndx 17378  df-base 17394  df-ress 17415  df-plusg 17447  df-hom 17458  df-cco 17459  df-0g 17618  df-cat 17848  df-cid 17849  df-homf 17850  df-ssc 17991  df-resc 17992  df-subc 17993  df-func 18039  df-resf 18042  df-setc 18257  df-estrc 18303  df-mgm 18822  df-mgmhm 18887  df-sgrp 18914  df-mnd 18930  df-mhm 18984  df-grp 19153  df-ghm 19434  df-abl 20003  df-mgp 20367  df-rng 20381  df-rnghm 20672  df-rngc 20875

This theorem is used by: (None)