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Theorem estrcval 18079

Description: Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)

**Hypotheses**
Ref	Expression
estrcval.c	⊢ 𝐶 = (ExtStrCat‘𝑈)
estrcval.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
estrcval.h	⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))))
estrcval.o	⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))

**Assertion**
Ref	Expression
estrcval	⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Distinct variable groups: 𝑓,𝑔,𝑣,𝑥,𝑦,𝑧 𝜑,𝑣,𝑥,𝑦,𝑧 𝑣,𝑈,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑓,𝑔) 𝐶(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔) · (𝑥,𝑦,𝑧,𝑣,𝑓,𝑔) 𝑈(𝑓,𝑔) 𝐻(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔) 𝑉(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)

Proof of Theorem estrcval Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		estrcval.c	. 2 ⊢ 𝐶 = (ExtStrCat‘𝑈)
2		df-estrc 18078	. . 3 ⊢ ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩})
3		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
4	3	opeq2d 4879	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Base‘ndx), 𝑢⟩ = ⟨(Base‘ndx), 𝑈⟩)
5		eqidd 2731	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((Base‘𝑦) ↑_m (Base‘𝑥)) = ((Base‘𝑦) ↑_m (Base‘𝑥)))
6	3, 3, 5	mpoeq123dv 7486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))))
7		estrcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))))
8	7	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))))
9	6, 8	eqtr4d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥))) = 𝐻)
10	9	opeq2d 4879	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥)))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
11	3	sqxpeqd 5707	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 × 𝑢) = (𝑈 × 𝑈))
12		eqidd 2731	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))
13	11, 3, 12	mpoeq123dv 7486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
14		estrcval.o	. . . . . . 7 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
15	14	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
16	13, 15	eqtr4d 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))) = · )
17	16	opeq2d 4879	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
18	4, 10, 17	tpeq123d 4751	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑_m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑_m (Base‘(2^nd ‘𝑣))), 𝑓 ∈ ((Base‘(2^nd ‘𝑣)) ↑_m (Base‘(1^st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
19		estrcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
20	19	elexd 3493	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
21		tpex 7736	. . . 4 ⊢ {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
22	21	a1i 11	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V)
23	2, 18, 20, 22	fvmptd2 7005	. 2 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
24	1, 23	eqtrid 2782	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 {ctp 4631 ⟨cop 4633 × cxp 5673 ∘ ccom 5679 ‘cfv 6542 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7975 2^nd c2nd 7976 ↑_m cmap 8822 ndxcnx 17130 Basecbs 17148 Hom chom 17212 compcco 17213 ExtStrCatcestrc 18077

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-oprab 7415 df-mpo 7416 df-estrc 18078

This theorem is referenced by: estrcbas 18080 estrchomfval 18081 estrccofval 18084 dfrngc2 46958 dfringc2 47004

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