Theorem funcrngcsetcALT 20601

Description: Alternate proof of funcrngcsetc 20600, using cofuval2 17900 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 20599, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18161. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 20600. (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 6876	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
3	2	cbvmptv 5225	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢))
4	1, 3	eqtrdi 2786	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
5		coires1 6253	. . . . . . 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 20581	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
10	9	eleq2d 2820	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng)))
11		elin 3942	. . . . . . . . . . 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 3964	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
15	14	resmptd 6027	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ↾ 𝐵) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
16	5, 15	eqtr2id 2783	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)) = ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
17	4, 16	eqtrd 2770	. . . . 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 6253	. . . . . . . . 9 ⊢ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦))) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHom 𝑦))
20		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘𝑥) = (Base‘𝑥)
21		eqid 2735	. . . . . . . . . . . . 13 ⊢ (Base‘𝑦) = (Base‘𝑦)
22	20, 21	rnghmf 20408	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑥 RngHom 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23		fvex 6889	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑦) ∈ V
24		fvex 6889	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑥) ∈ V
25	23, 24	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26		elmapg 8853	. . . . . . . . . . . . 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 3964	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 RngHom 𝑦) ⊆ ((Base‘𝑦) ↑m (Base‘𝑥)))
30	29	resabs1d 5995	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑥 RngHom 𝑦)))
31	19, 30	eqtr2id 2783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ( I ↾ (𝑥 RngHom 𝑦)) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦))))
32	31	mpoeq3dva 7484	. . . . . . 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 2770	. . . . . 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 7165	. . . . . . . . . . . 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 7165	. . . . . . . . . . . 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 7423	. . . . . . . . 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 2736	. . . . . . . . . 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 6880	. . . . . . . . . . . 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 6880	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
48	45, 47	oveq12d 7423	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑦) ↑m (Base‘𝑥)))
49	48	reseq2d 5966	. . . . . . . . . 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 2820	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (𝑈 ∩ Rng)))
54		elin 3942	. . . . . . . . . . . . . 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 7438	. . . . . . . . . . . 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 7208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
62	43, 49, 52, 58, 61	ovmpod 7559	. . . . . . . . 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 2771	. . . . . . . 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 2736	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))))
65		oveq12 7414	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝑥 ∧ 𝑔 = 𝑦) → (𝑓 RngHom 𝑔) = (𝑥 RngHom 𝑦))
66	65	reseq2d 5966	. . . . . . . . . . 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 7438	. . . . . . . . . . . 12 ⊢ (𝑥 RngHom 𝑦) ∈ V
71	70	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) ∈ V)
72	71	resiexd 7208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) ∈ V)
73	64, 67, 68, 69, 72	ovmpod 7559	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦) = ( I ↾ (𝑥 RngHom 𝑦)))
74	73	eqcomd 2741	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) = (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦))
75	63, 74	coeq12d 5844	. . . . . . 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 7481	. . . . . 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 2770	. . . . 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 4857	. . . 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 2735	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
80		eqid 2735	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
81		eqidd 2736	. . . . . 6 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
82		eqidd 2736	. . . . . 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 20599	. . . . 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 2735	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
86		eqid 2735	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
87	80, 8	estrcbas 18137	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
88	87	mpteq1d 5210	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89		fveq2 6876	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
90	89	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑧) ↑m (Base‘𝑢)))
91	90	reseq2d 5966	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))))
92		fveq2 6876	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
93	92	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝑧 = 𝑣 → ((Base‘𝑧) ↑m (Base‘𝑢)) = ((Base‘𝑣) ↑m (Base‘𝑢)))
94	93	reseq2d 5966	. . . . . . . . 9 ⊢ (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
95	91, 94	cbvmpov 7502	. . . . . . . 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 2736	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
98	87, 87, 97	mpoeq123dv 7482	. . . . . . 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 2770	. . . . . 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 18161	. . . . 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 17900	. . . 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 2773	. . 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 5120	. . . . 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 5120	. . . . 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 17901	. . 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 2834	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
109		df-br 5120	. 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 1540   ∈ wcel 2108  Vcvv 3459   ∩ cin 3925  ⟨cop 4607   class class class wbr 5119   ↦ cmpt 5201   I cid 5547   ↾ cres 5656   ∘ ccom 5658  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407   ↑_m cmap 8840  WUnicwun 10714  Basecbs 17228   Func cfunc 17867   ∘_func ccofu 17869  SetCatcsetc 18088  ExtStrCatcestrc 18134  Rngcrng 20112   RngHom crnghm 20394  RngCatcrngc 20576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-pm 8843  df-ixp 8912  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-wun 10716  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-struct 17166  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17252  df-plusg 17284  df-hom 17295  df-cco 17296  df-0g 17455  df-cat 17680  df-cid 17681  df-homf 17682  df-ssc 17823  df-resc 17824  df-subc 17825  df-func 17871  df-idfu 17872  df-cofu 17873  df-full 17919  df-fth 17920  df-setc 18089  df-estrc 18135  df-mgm 18618  df-mgmhm 18670  df-sgrp 18697  df-mnd 18713  df-mhm 18761  df-grp 18919  df-ghm 19196  df-abl 19764  df-mgp 20101  df-rng 20113  df-rnghm 20396  df-rngc 20577

This theorem is referenced by: (None)