Theorem funcrngcsetcALT 20574

Description: Alternate proof of funcrngcsetc 20573, using cofuval2 17811 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 20572, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18072. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 20573. (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 6834	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
3	2	cbvmptv 5202	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢))
4	1, 3	eqtrdi 2787	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
5		coires1 6223	. . . . . . 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 20554	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
10	9	eleq2d 2822	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng)))
11		elin 3917	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng))
12	11	simplbi 497	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥 ∈ 𝑈)
13	10, 12	biimtrdi 253	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈))
14	13	ssrdv 3939	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
15	14	resmptd 5999	. . . . . . 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 6223	. . . . . . . . 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 20384	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑥 RngHom 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23		fvex 6847	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑦) ∈ V
24		fvex 6847	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑥) ∈ V
25	23, 24	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26		elmapg 8776	. . . . . . . . . . . . 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 3939	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 RngHom 𝑦) ⊆ ((Base‘𝑦) ↑m (Base‘𝑥)))
30	29	resabs1d 5967	. . . . . . . . 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 7435	. . . . . . 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 7119	. . . . . . . . . . . 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 7119	. . . . . . . . . . . 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 7376	. . . . . . . . 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 772	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → 𝑧 = 𝑦)
45	44	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → (Base‘𝑧) = (Base‘𝑦))
46		simprl 770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → 𝑤 = 𝑥)
47	46	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
48	45, 47	oveq12d 7376	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑦) ↑m (Base‘𝑥)))
49	48	reseq2d 5938	. . . . . . . . . 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 3917	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝑈 ∩ Rng) ↔ (𝑦 ∈ 𝑈 ∧ 𝑦 ∈ Rng))
55	54	simplbi 497	. . . . . . . . . . . . 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 7391	. . . . . . . . . . . 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 7162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
62	43, 49, 52, 58, 61	ovmpod 7510	. . . . . . . . 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 7367	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝑥 ∧ 𝑔 = 𝑦) → (𝑓 RngHom 𝑔) = (𝑥 RngHom 𝑦))
66	65	reseq2d 5938	. . . . . . . . . . 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 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
69		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
70		ovex 7391	. . . . . . . . . . . 12 ⊢ (𝑥 RngHom 𝑦) ∈ V
71	70	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) ∈ V)
72	71	resiexd 7162	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) ∈ V)
73	64, 67, 68, 69, 72	ovmpod 7510	. . . . . . . . 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 5813	. . . . . . 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 7432	. . . . . 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 4837	. . . 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 20572	. . . . 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 18048	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
88	87	mpteq1d 5188	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
90	89	oveq2d 7374	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑧) ↑m (Base‘𝑢)))
91	90	reseq2d 5938	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))))
92		fveq2 6834	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
93	92	oveq1d 7373	. . . . . . . . . 10 ⊢ (𝑧 = 𝑣 → ((Base‘𝑧) ↑m (Base‘𝑢)) = ((Base‘𝑣) ↑m (Base‘𝑢)))
94	93	reseq2d 5938	. . . . . . . . 9 ⊢ (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
95	91, 94	cbvmpov 7453	. . . . . . . 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 7433	. . . . . . 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 18072	. . . . 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 17811	. . . 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 5099	. . . . 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 5099	. . . . 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 17812	. . 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 5099	. 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 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∩ cin 3900  ⟨cop 4586   class class class wbr 5098   ↦ cmpt 5179   I cid 5518   ↾ cres 5626   ∘ ccom 5628  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360   ↑_m cmap 8763  WUnicwun 10611  Basecbs 17136   Func cfunc 17778   ∘_func ccofu 17780  SetCatcsetc 17999  ExtStrCatcestrc 18045  Rngcrng 20087   RngHom crnghm 20370  RngCatcrngc 20549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  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 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-er 8635  df-map 8765  df-pm 8766  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-wun 10613  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-hom 17201  df-cco 17202  df-0g 17361  df-cat 17591  df-cid 17592  df-homf 17593  df-ssc 17734  df-resc 17735  df-subc 17736  df-func 17782  df-idfu 17783  df-cofu 17784  df-full 17830  df-fth 17831  df-setc 18000  df-estrc 18046  df-mgm 18565  df-mgmhm 18617  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-grp 18866  df-ghm 19142  df-abl 19712  df-mgp 20076  df-rng 20088  df-rnghm 20372  df-rngc 20550

This theorem is referenced by: (None)