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Theorem setcval 18009

Description: Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

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

**Assertion**
Ref	Expression
setcval	⊢ (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 setcval Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		setcval.c	. 2 ⊢ 𝐶 = (SetCat‘𝑈)
2		df-setc 18008	. . 3 ⊢ SetCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑_m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
3		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
4	3	opeq2d 4873	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Base‘ndx), 𝑢⟩ = ⟨(Base‘ndx), 𝑈⟩)
5		eqidd 2732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑦 ↑_m 𝑥) = (𝑦 ↑_m 𝑥))
6	3, 3, 5	mpoeq123dv 7468	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑_m 𝑥)) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑_m 𝑥)))
7		setcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑_m 𝑥)))
8	7	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ (𝑦 ↑_m 𝑥)))
9	6, 8	eqtr4d 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑_m 𝑥)) = 𝐻)
10	9	opeq2d 4873	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑_m 𝑥))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
11	3	sqxpeqd 5701	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 × 𝑢) = (𝑈 × 𝑈))
12		eqidd 2732	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
13	11, 3, 12	mpoeq123dv 7468	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
14		setcval.o	. . . . . . 7 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
16	13, 15	eqtr4d 2774	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = · )
17	16	opeq2d 4873	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
18	4, 10, 17	tpeq123d 4745	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑_m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑_m (2^nd ‘𝑣)), 𝑓 ∈ ((2^nd ‘𝑣) ↑_m (1^st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
19		setcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
20	19	elexd 3493	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
21		tpex 7717	. . . 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 6992	. 2 ⊢ (𝜑 → (SetCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
24	1, 23	eqtrid 2783	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 {ctp 4626 ⟨cop 4628 × cxp 5667 ∘ ccom 5673 ‘cfv 6532 (class class class)co 7393 ∈ cmpo 7395 1^st c1st 7955 2^nd c2nd 7956 ↑_m cmap 8803 ndxcnx 17108 Basecbs 17126 Hom chom 17190 compcco 17191 SetCatcsetc 18007

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6484 df-fun 6534 df-fv 6540 df-oprab 7397 df-mpo 7398 df-setc 18008

This theorem is referenced by: setcbas 18010 setchomfval 18011 setccofval 18014

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