Theorem funcrngcsetcALT 20572

Description: Alternate proof of funcrngcsetc 20571, using cofuval2 17809 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 20570, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 18070. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 20571. (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 6832	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (Base‘𝑥) = (Base‘𝑢))
3	2	cbvmptv 5200	. . . . . . 7 ⊢ (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢))
4	1, 3	eqtrdi 2785	. . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
5		coires1 6221	. . . . . . 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 20552	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
10	9	eleq2d 2820	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (𝑈 ∩ Rng)))
11		elin 3915	. . . . . . . . . . 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 3937	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑈)
15	14	resmptd 5997	. . . . . . 7 ⊢ (𝜑 → ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ↾ 𝐵) = (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)))
16	5, 15	eqtr2id 2782	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝐵 ↦ (Base‘𝑢)) = ((𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) ∘ ( I ↾ 𝐵)))
17	4, 16	eqtrd 2769	. . . . 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 6221	. . . . . . . . 9 ⊢ (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦))) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHom 𝑦))
20		eqid 2734	. . . . . . . . . . . . 13 ⊢ (Base‘𝑥) = (Base‘𝑥)
21		eqid 2734	. . . . . . . . . . . . 13 ⊢ (Base‘𝑦) = (Base‘𝑦)
22	20, 21	rnghmf 20382	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑥 RngHom 𝑦) → 𝑧:(Base‘𝑥)⟶(Base‘𝑦))
23		fvex 6845	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑦) ∈ V
24		fvex 6845	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑥) ∈ V
25	23, 24	pm3.2i 470	. . . . . . . . . . . . 13 ⊢ ((Base‘𝑦) ∈ V ∧ (Base‘𝑥) ∈ V)
26		elmapg 8774	. . . . . . . . . . . . 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 3937	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 RngHom 𝑦) ⊆ ((Base‘𝑦) ↑m (Base‘𝑥)))
30	29	resabs1d 5965	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ↾ (𝑥 RngHom 𝑦)) = ( I ↾ (𝑥 RngHom 𝑦)))
31	19, 30	eqtr2id 2782	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ( I ↾ (𝑥 RngHom 𝑦)) = (( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∘ ( I ↾ (𝑥 RngHom 𝑦))))
32	31	mpoeq3dva 7433	. . . . . . 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 2769	. . . . . 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 7117	. . . . . . . . . . . 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 7117	. . . . . . . . . . . 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 7374	. . . . . . . . 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 2735	. . . . . . . . . 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 6836	. . . . . . . . . . . 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 6836	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → (Base‘𝑤) = (Base‘𝑥))
48	45, 47	oveq12d 7374	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑤 = 𝑥 ∧ 𝑧 = 𝑦)) → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑦) ↑m (Base‘𝑥)))
49	48	reseq2d 5936	. . . . . . . . . 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 3915	. . . . . . . . . . . . . 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 7389	. . . . . . . . . . . 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 7160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ ((Base‘𝑦) ↑m (Base‘𝑥))) ∈ V)
62	43, 49, 52, 58, 61	ovmpod 7508	. . . . . . . . 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 2770	. . . . . . . 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 2735	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔))))
65		oveq12 7365	. . . . . . . . . . . 12 ⊢ ((𝑓 = 𝑥 ∧ 𝑔 = 𝑦) → (𝑓 RngHom 𝑔) = (𝑥 RngHom 𝑦))
66	65	reseq2d 5936	. . . . . . . . . . 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 7389	. . . . . . . . . . . 12 ⊢ (𝑥 RngHom 𝑦) ∈ V
71	70	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 RngHom 𝑦) ∈ V)
72	71	resiexd 7160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) ∈ V)
73	64, 67, 68, 69, 72	ovmpod 7508	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦) = ( I ↾ (𝑥 RngHom 𝑦)))
74	73	eqcomd 2740	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( I ↾ (𝑥 RngHom 𝑦)) = (𝑥(𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ ( I ↾ (𝑓 RngHom 𝑔)))𝑦))
75	63, 74	coeq12d 5811	. . . . . . 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 7430	. . . . . 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 2769	. . . . 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 4835	. . . 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 2734	. . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅)
80		eqid 2734	. . . . . 6 ⊢ (ExtStrCat‘𝑈) = (ExtStrCat‘𝑈)
81		eqidd 2735	. . . . . 6 ⊢ (𝜑 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
82		eqidd 2735	. . . . . 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 20570	. . . . 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 2734	. . . . . 6 ⊢ (Base‘(ExtStrCat‘𝑈)) = (Base‘(ExtStrCat‘𝑈))
86		eqid 2734	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
87	80, 8	estrcbas 18046	. . . . . . 7 ⊢ (𝜑 → 𝑈 = (Base‘(ExtStrCat‘𝑈)))
88	87	mpteq1d 5186	. . . . . 6 ⊢ (𝜑 → (𝑢 ∈ 𝑈 ↦ (Base‘𝑢)) = (𝑢 ∈ (Base‘(ExtStrCat‘𝑈)) ↦ (Base‘𝑢)))
89		fveq2 6832	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑢 → (Base‘𝑤) = (Base‘𝑢))
90	89	oveq2d 7372	. . . . . . . . . 10 ⊢ (𝑤 = 𝑢 → ((Base‘𝑧) ↑m (Base‘𝑤)) = ((Base‘𝑧) ↑m (Base‘𝑢)))
91	90	reseq2d 5936	. . . . . . . . 9 ⊢ (𝑤 = 𝑢 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑤))) = ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))))
92		fveq2 6832	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑣 → (Base‘𝑧) = (Base‘𝑣))
93	92	oveq1d 7371	. . . . . . . . . 10 ⊢ (𝑧 = 𝑣 → ((Base‘𝑧) ↑m (Base‘𝑢)) = ((Base‘𝑣) ↑m (Base‘𝑢)))
94	93	reseq2d 5936	. . . . . . . . 9 ⊢ (𝑧 = 𝑣 → ( I ↾ ((Base‘𝑧) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
95	91, 94	cbvmpov 7451	. . . . . . . 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 2735	. . . . . . . 8 ⊢ (𝜑 → ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))) = ( I ↾ ((Base‘𝑣) ↑m (Base‘𝑢))))
98	87, 87, 97	mpoeq123dv 7431	. . . . . . 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 2769	. . . . . 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 18070	. . . . 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 17809	. . . 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 2772	. . 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 5097	. . . . 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 5097	. . . . 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 17810	. . 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 5097	. 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 3438   ∩ cin 3898  ⟨cop 4584   class class class wbr 5096   ↦ cmpt 5177   I cid 5516   ↾ cres 5624   ∘ ccom 5626  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358   ↑_m cmap 8761  WUnicwun 10609  Basecbs 17134   Func cfunc 17776   ∘_func ccofu 17778  SetCatcsetc 17997  ExtStrCatcestrc 18043  Rngcrng 20085   RngHom crnghm 20368  RngCatcrngc 20547

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 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-map 8763  df-pm 8764  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-wun 10611  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-fz 13422  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-hom 17199  df-cco 17200  df-0g 17359  df-cat 17589  df-cid 17590  df-homf 17591  df-ssc 17732  df-resc 17733  df-subc 17734  df-func 17780  df-idfu 17781  df-cofu 17782  df-full 17828  df-fth 17829  df-setc 17998  df-estrc 18044  df-mgm 18563  df-mgmhm 18615  df-sgrp 18642  df-mnd 18658  df-mhm 18706  df-grp 18864  df-ghm 19140  df-abl 19710  df-mgp 20074  df-rng 20086  df-rnghm 20370  df-rngc 20548

This theorem is referenced by: (None)