Theorem funcrngcsetcALT 44623

Description: Alternate proof of funcrngcsetc 44622, using cofuval2 17149 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 44621, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 17391. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 44622. (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 RngHomo 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 6645	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
3	2	cbvmptv 5133	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢))
4	1, 3	eqtrdi 2849	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
5		coires1 6084	. . . . . . 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 44589	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
10	9	eleq2d 2875	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng)))
11		elin 3897	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) ↔ (𝑥 ∈ 𝑈 ∧ 𝑥 ∈ Rng))
12	11	simplbi 501	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑈 ∩ Rng) → 𝑥 ∈ 𝑈)
13	10, 12	syl6bi 256	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝑈))
14	13	ssrdv 3921	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
15	14	resmptd 5875	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ↾ 𝐵) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
16	5, 15	syl5req 2846	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)) = ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
17	4, 16	eqtrd 2833	. . . . 5 ⊢ (𝜑 → 𝐹 = ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
18		funcrngcsetcALT.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))
19		coires1 6084	. . . . . . . . 9 ⊢ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦))
20		eqid 2798	. . . . . . . . . . . . 13 ⊢ (Base‘𝑥) = (Base‘𝑥)
21		eqid 2798	. . . . . . . . . . . . 13 ⊢ (Base‘𝑦) = (Base‘𝑦)
22	20, 21	rnghmf 44523	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23		fvex 6658	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑦) ∈ V
24		fvex 6658	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑥) ∈ V
25	23, 24	pm3.2i 474	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26		elmapg 8402	. . . . . . . . . . . . 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	syl5ibr 249	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 ∈ (𝑥 RngHomo 𝑦) → 𝑧 ∈ ((Base‘𝑦) ↑m (Base‘𝑥))))
29	28	ssrdv 3921	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 RngHomo 𝑦) ⊆ ((Base‘𝑦) ↑m (Base‘𝑥)))
30	29	resabs1d 5849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHomo 𝑦)) = ( I ↾ (𝑥 RngHomo 𝑦)))
31	19, 30	syl5req 2846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ( I ↾ (𝑥 RngHomo 𝑦)) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))))
32	31	mpoeq3dva 7210	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦)))))
33	18, 32	eqtrd 2833	. . . . . 6 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦)))))
34	7	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
35	7	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐵 = (Base‘𝑅))
36		fvresi 6912	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑥) = 𝑥)
37	36	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
38	37	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑥) = 𝑥)
39		fvresi 6912	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝐵 → (( I ↾ 𝐵)‘𝑦) = 𝑦)
40	39	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
41	40	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (( I ↾ 𝐵)‘𝑦) = 𝑦)
42	38, 41	oveq12d 7153	. . . . . . . . 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 2799	. . . . . . . . . 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 6649	. . . . . . . . . . . 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 6649	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
48	45, 47	oveq12d 7153	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑦) ↑m (Base‘𝑥)))
49	48	reseq2d 5818	. . . . . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝜑 → 𝑥 ∈ 𝑈))
52	51	impcom 411	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝑈)
53	9	eleq2d 2875	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ (𝑈 ∩ Rng)))
54		elin 3897	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝑈 ∩ Rng) ↔ (𝑦 ∈ 𝑈 ∧ 𝑦 ∈ Rng))
55	54	simplbi 501	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑈 ∩ Rng) → 𝑦 ∈ 𝑈)
56	53, 55	syl6bi 256	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑈))
57	56	a1d 25	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝑈)))
58	57	imp32 422	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝑈)
59		ovex 7168	. . . . . . . . . . . 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 6956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
62	43, 49, 52, 58, 61	ovmpod 7281	. . . . . . . . 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 2834	. . . . . . . 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 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
65		oveq12 7144	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝑥 ∧ 𝑔 = 𝑦) → (𝑓 RngHomo 𝑔) = (𝑥 RngHomo 𝑦))
66	65	reseq2d 5818	. . . . . . . . . . 11 ⊢ ((𝑓 = 𝑥 ∧ 𝑔 = 𝑦) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 𝑦)))
67	66	adantl 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑓 = 𝑥 ∧ 𝑔 = 𝑦)) → ( I ↾ (𝑓 RngHomo 𝑔)) = ( I ↾ (𝑥 RngHomo 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 7168	. . . . . . . . . . . 12 ⊢ (𝑥 RngHomo 𝑦) ∈ V
71	70	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHomo 𝑦) ∈ V)
72	71	resiexd 6956	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) ∈ V)
73	64, 67, 68, 69, 72	ovmpod 7281	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦) = ( I ↾ (𝑥 RngHomo 𝑦)))
74	73	eqcomd 2804	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHomo 𝑦)) = (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦))
75	63, 74	coeq12d 5699	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 𝑦))) = (((( I ↾ 𝐵)‘𝑥)(𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))(( I ↾ 𝐵)‘𝑦)) ∘ (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))𝑦)))
76	34, 35, 75	mpoeq123dva 7207	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHomo 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 RngHomo 𝑔)))𝑦))))
77	33, 76	eqtrd 2833	. . . . 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 RngHomo 𝑔)))𝑦))))
78	17, 77	opeq12d 4773	. . . 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 RngHomo 𝑔)))𝑦)))⟩)
79		eqid 2798	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
80		eqid 2798	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
81		eqidd 2799	. . . . . 6 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
82		eqidd 2799	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
83	6, 80, 7, 8, 81, 82	rngcifuestrc 44621	. . . . 5 ⊢ (𝜑 → ( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))))
84		funcrngcsetcALT.s	. . . . . 6 ⊢ 𝑆 = (SetCat‘𝑈)
85		eqid 2798	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
86		eqid 2798	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
87	80, 8	estrcbas 17367	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
88	87	mpteq1d 5119	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
90	89	oveq2d 7151	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑧) ↑m (Base‘𝑢)))
91	90	reseq2d 5818	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))))
92		fveq2 6645	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
93	92	oveq1d 7150	. . . . . . . . . 10 ⊢ (𝑧 = 𝑣 → ((Base‘𝑧) ↑m (Base‘𝑢)) = ((Base‘𝑣) ↑m (Base‘𝑢)))
94	93	reseq2d 5818	. . . . . . . . 9 ⊢ (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
95	91, 94	cbvmpov 7228	. . . . . . . 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 2799	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
98	87, 87, 97	mpoeq123dv 7208	. . . . . . 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 2833	. . . . . 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 17391	. . . . 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 17149	. . . 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 RngHomo 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 RngHomo 𝑔)))𝑦)))⟩)
102	78, 101	eqtr4d 2836	. . 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 RngHomo 𝑔)))⟩))
103		df-br 5031	. . . . 5 ⊢ (( I ↾ 𝐵)(𝑅 Func (ExtStrCat‘𝑈))(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔))) ↔ ⟨( I ↾ 𝐵), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
104	83, 103	sylib 221	. . . 4 ⊢ (𝜑 → ⟨( I ↾ 𝐵), (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHomo 𝑔)))⟩ ∈ (𝑅 Func (ExtStrCat‘𝑈)))
105		df-br 5031	. . . . 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 221	. . . 4 ⊢ (𝜑 → ⟨(𝑢 ∈ 𝑈 ↦ (Base‘𝑢)), (𝑤 ∈ 𝑈, 𝑧 ∈ 𝑈 ↦ ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))))⟩ ∈ ((ExtStrCat‘𝑈) Func 𝑆))
107	104, 106	cofucl 17150	. . 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 RngHomo 𝑔)))⟩) ∈ (𝑅 Func 𝑆))
108	102, 107	eqeltrd 2890	. 2 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
109		df-br 5031	. 2 ⊢ (𝐹(𝑅 Func 𝑆)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑅 Func 𝑆))
110	108, 109	sylibr 237	1 ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∩ cin 3880  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110   I cid 5424   ↾ cres 5521   ∘ ccom 5523  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137   ↑_m cmap 8389  WUnicwun 10111  Basecbs 16475   Func cfunc 17116   ∘_func ccofu 17118  SetCatcsetc 17327  ExtStrCatcestrc 17364  Rngcrng 44498   RngHomo crngh 44509  RngCatcrngc 44581

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-pm 8392  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-wun 10113  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-hom 16581  df-cco 16582  df-0g 16707  df-cat 16931  df-cid 16932  df-homf 16933  df-ssc 17072  df-resc 17073  df-subc 17074  df-func 17120  df-idfu 17121  df-cofu 17122  df-full 17166  df-fth 17167  df-setc 17328  df-estrc 17365  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-mhm 17948  df-grp 18098  df-ghm 18348  df-abl 18901  df-mgp 19233  df-mgmhm 44399  df-rng0 44499  df-rnghomo 44511  df-rngc 44583

This theorem is referenced by: (None)