setcval - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > setcval		Structured version Visualization version GIF version

Theorem setcval 18032

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 18031	. . 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 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
4	3	opeq2d 4880	. . . 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 7487	. . . . . 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 480	. . . . . 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 4880	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑_m 𝑥))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
11	3	sqxpeqd 5708	. . . . . . 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 7487	. . . . . 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 480	. . . . . 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 4880	. . . 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 4752	. . 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 3494	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
21		tpex 7737	. . . 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 7006	. 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 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 {ctp 4632 ⟨cop 4634 × cxp 5674 ∘ ccom 5680 ‘cfv 6543 (class class class)co 7412 ∈ cmpo 7414 1^st c1st 7976 2^nd c2nd 7977 ↑_m cmap 8823 ndxcnx 17131 Basecbs 17149 Hom chom 17213 compcco 17214 SetCatcsetc 18030

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7728

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-oprab 7416 df-mpo 7417 df-setc 18031

This theorem is referenced by: setcbas 18033 setchomfval 18034 setccofval 18037

W3C validator