Theorem funcrngcsetcALT 20618

Description: Alternate proof of funcrngcsetc 20617, using cofuval2 17854 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 20616, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18115. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 20617. (Contributed by AV, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem funcrngcsetcALT

Dummy variables 𝑓 𝑔 𝑢 𝑤 𝑧 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funcrngcsetcALT.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))
2		fveq2 6840	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
3	2	cbvmptv 5189	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢))
4	1, 3	eqtrdi 2787	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
5		coires1 6229	. . . . . . 7 ⊢ ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)) = ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ↾ 𝐵)
6		funcrngcsetcALT.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (RngCat‘𝑈)
7		funcrngcsetcALT.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑅)
8		funcrngcsetcALT.u	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ WUni)
9	6, 7, 8	rngcbas 20598	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
10	9	eleq2d 2822	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng)))
11		elin 3905	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng))
12	11	simplbi 496	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥 ∈ 𝑈)
13	10, 12	biimtrdi 253	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈))
14	13	ssrdv 3927	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
15	14	resmptd 6005	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ↾ 𝐵) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
16	5, 15	eqtr2id 2784	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)) = ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
17	4, 16	eqtrd 2771	. . . . 5 ⊢ (𝜑 → 𝐹 = ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
18		funcrngcsetcALT.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))))
19		coires1 6229	. . . . . . . . 9 ⊢ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦))) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHom 𝑦))
20		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝑥) = (Base‘𝑥)
21		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Base‘𝑦) = (Base‘𝑦)
22	20, 21	rnghmf 20428	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑥 RngHom 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23		fvex 6853	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑦) ∈ V
24		fvex 6853	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑥) ∈ V
25	23, 24	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26		elmapg 8786	. . . . . . . . . . . . 13 ⊢ (((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V) → (𝑧 ∈ ((Base‘𝑦) ↑m (Base‘𝑥)) ↔ 𝑧:(Base‘𝑥)⟶(Base‘𝑦)))
27	25, 26	mp1i 13	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ ((Base‘𝑦) ↑m (Base‘𝑥)) ↔ 𝑧:(Base‘𝑥)⟶(Base‘𝑦)))
28	22, 27	imbitrrid 246	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ (𝑥 RngHom 𝑦) → 𝑧 ∈ ((Base‘𝑦) ↑m (Base‘𝑥))))
29	28	ssrdv 3927	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 RngHom 𝑦) ⊆ ((Base‘𝑦) ↑m (Base‘𝑥)))
30	29	resabs1d 5973	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑥 RngHom 𝑦)))
31	19, 30	eqtr2id 2784	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ( I ↾ (𝑥 RngHom 𝑦)) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦))))
32	31	mpoeq3dva 7444	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHom 𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦)))))
33	18, 32	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦)))))
34	7	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
35	7	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘𝑅))
36		fvresi 7128	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
37	36	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
38	37	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
39		fvresi 7128	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
40	39	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
41	40	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
42	38, 41	oveq12d 7385	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((( I ↾ 𝐵)‘𝑥)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) = (𝑥(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))𝑦))
43		eqidd 2737	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))) = (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))))
44		simprr 773	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → 𝑧 = 𝑦)
45	44	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → (Base‘𝑧) = (Base‘𝑦))
46		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → 𝑤 = 𝑥)
47	46	fveq2d 6844	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
48	45, 47	oveq12d 7385	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑦) ↑m (Base‘𝑥)))
49	48	reseq2d 5944	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))
50	13	com12 32	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → (𝜑 → 𝑥 ∈ 𝑈))
51	50	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝑥 ∈ 𝑈))
52	51	impcom 407	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑈)
53	9	eleq2d 2822	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (𝑈 ∩ Rng)))
54		elin 3905	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝑈 ∩ Rng) ↔ (𝑦 ∈ 𝑈 ∧ 𝑦 ∈ Rng))
55	54	simplbi 496	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑈 ∩ Rng) → 𝑦 ∈ 𝑈)
56	53, 55	biimtrdi 253	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑈))
57	56	a1d 25	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑈)))
58	57	imp32 418	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝑈)
59		ovex 7400	. . . . . . . . . . . 12 ⊢ ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V
60	59	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((Base‘𝑦) ↑m (Base‘𝑥)) ∈ V)
61	60	resiexd 7171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
62	43, 49, 52, 58, 61	ovmpod 7519	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))𝑦) = ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))))
63	42, 62	eqtr2d 2772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) = ((( I ↾ 𝐵)‘𝑥)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)))
64		eqidd 2737	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))))
65		oveq12 7376	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝑥 ∧ 𝑔 = 𝑦) → (𝑓 RngHom 𝑔) = (𝑥 RngHom 𝑦))
66	65	reseq2d 5944	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝑥 ∧ 𝑔 = 𝑦) → ( I ↾ (𝑓 RngHom 𝑔)) = ( I ↾ (𝑥 RngHom 𝑦)))
67	66	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑓 = 𝑥 ∧ 𝑔 = 𝑦)) → ( I ↾ (𝑓 RngHom 𝑔)) = ( I ↾ (𝑥 RngHom 𝑦)))
68		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
69		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
70		ovex 7400	. . . . . . . . . . . 12 ⊢ (𝑥 RngHom 𝑦) ∈ V
71	70	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) ∈ V)
72	71	resiexd 7171	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) ∈ V)
73	64, 67, 68, 69, 72	ovmpod 7519	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦) = ( I ↾ (𝑥 RngHom 𝑦)))
74	73	eqcomd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) = (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦))
75	63, 74	coeq12d 5819	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦))) = (((( I ↾ 𝐵)‘𝑥)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦)))
76	34, 35, 75	mpoeq123dva 7441	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦)))) = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (((( I ↾ 𝐵)‘𝑥)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦))))
77	33, 76	eqtrd 2771	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (((( I ↾ 𝐵)‘𝑥)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦))))
78	17, 77	opeq12d 4824	. . . 4 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = ⟨((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (((( I ↾ 𝐵)‘𝑥)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦)))⟩)
79		eqid 2736	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
80		eqid 2736	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
81		eqidd 2737	. . . . . 6 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
82		eqidd 2737	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))))
83	6, 80, 7, 8, 81, 82	rngcifuestrc 20616	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))))
84		funcrngcsetcALT.s	. . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈)
85		eqid 2736	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
86		eqid 2736	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
87	80, 8	estrcbas 18091	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
88	87	mpteq1d 5175	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
90	89	oveq2d 7383	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑧) ↑m (Base‘𝑢)))
91	90	reseq2d 5944	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))))
92		fveq2 6840	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
93	92	oveq1d 7382	. . . . . . . . . 10 ⊢ (𝑧 = 𝑣 → ((Base‘𝑧) ↑m (Base‘𝑢)) = ((Base‘𝑣) ↑m (Base‘𝑢)))
94	93	reseq2d 5944	. . . . . . . . 9 ⊢ (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
95	91, 94	cbvmpov 7462	. . . . . . . 8 ⊢ (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))) = (𝑢 ∈ 𝑈, 𝑣 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
96	95	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))) = (𝑢 ∈ 𝑈, 𝑣 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢)))))
97		eqidd 2737	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
98	87, 87, 97	mpoeq123dv 7442	. . . . . . 7 ⊢ (𝜑 → (𝑢 ∈ 𝑈, 𝑣 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢)))) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑣 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢)))))
99	96, 98	eqtrd 2771	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)), 𝑣 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢)))))
100	80, 84, 85, 86, 8, 88, 99	funcestrcsetc 18115	. . . . 5 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↦ (Base‘𝑢))((ExtStrCat‘𝑈) Func 𝑆)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))))
101	79, 83, 100	cofuval2 17854	. . . 4 ⊢ (𝜑 → (⟨(𝑢 ∈ 𝑈 ↦ (Base‘𝑢)), (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))⟩ ∘func ⟨( I ↾ 𝐵), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))⟩) = ⟨((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ (Base‘𝑅) ↦ (((( I ↾ 𝐵)‘𝑥)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦)))⟩)
102	78, 101	eqtr4d 2774	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ = (⟨(𝑢 ∈ 𝑈 ↦ (Base‘𝑢)), (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))⟩ ∘func ⟨( I ↾ 𝐵), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))⟩))
103		df-br 5086	. . . . 5 ⊢ (( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))) ↔ ⟨( I ↾ 𝐵), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
104	83, 103	sylib 218	. . . 4 ⊢ (𝜑 → ⟨( I ↾ 𝐵), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
105		df-br 5086	. . . . 5 ⊢ ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢))((ExtStrCat‘𝑈) Func 𝑆)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤)))) ↔ ⟨(𝑢 ∈ 𝑈 ↦ (Base‘𝑢)), (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
106	100, 105	sylib 218	. . . 4 ⊢ (𝜑 → ⟨(𝑢 ∈ 𝑈 ↦ (Base‘𝑢)), (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
107	104, 106	cofucl 17855	. . 3 ⊢ (𝜑 → (⟨(𝑢 ∈ 𝑈 ↦ (Base‘𝑢)), (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))⟩ ∘func ⟨( I ↾ 𝐵), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))⟩) ∈ (𝑅 Func 𝑆))
108	102, 107	eqeltrd 2836	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
109		df-br 5086	. 2 ⊢ (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
110	108, 109	sylibr 234	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3429   ∩ cin 3888  ⟨cop 4573   class class class wbr 5085   ↦ cmpt 5166   I cid 5525   ↾ cres 5633   ∘ ccom 5635  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369   ↑_m cmap 8773  WUnicwun 10623  Basecbs 17179   Func cfunc 17821   ∘_func ccofu 17823  SetCatcsetc 18042  ExtStrCatcestrc 18088  Rngcrng 20133   RngHom crnghm 20414  RngCatcrngc 20593

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-er 8643  df-map 8775  df-pm 8776  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-wun 10625  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-fz 13462  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-hom 17244  df-cco 17245  df-0g 17404  df-cat 17634  df-cid 17635  df-homf 17636  df-ssc 17777  df-resc 17778  df-subc 17779  df-func 17825  df-idfu 17826  df-cofu 17827  df-full 17873  df-fth 17874  df-setc 18043  df-estrc 18089  df-mgm 18608  df-mgmhm 18660  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-grp 18912  df-ghm 19188  df-abl 19758  df-mgp 20122  df-rng 20134  df-rnghm 20416  df-rngc 20594

This theorem is referenced by: (None)