Theorem setcval 18045

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 (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘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 18044	. . 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 (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
3		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
4	3	opeq2d 4846	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Base‘ndx), 𝑢⟩ = ⟨(Base‘ndx), 𝑈⟩)
5		eqidd 2731	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑦 ↑m 𝑥) = (𝑦 ↑m 𝑥))
6	3, 3, 5	mpoeq123dv 7466	. . . . . 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 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑m 𝑥)) = 𝐻)
10	9	opeq2d 4846	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ (𝑦 ↑m 𝑥))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
11	3	sqxpeqd 5672	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 × 𝑢) = (𝑈 × 𝑈))
12		eqidd 2731	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))
13	11, 3, 12	mpoeq123dv 7466	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
14		setcval.o	. . . . . . 7 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
15	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))
16	13, 15	eqtr4d 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓))) = · )
17	16	opeq2d 4846	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ (𝑧 ↑m (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
18	4, 10, 17	tpeq123d 4714	. . 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 (2nd ‘𝑣)), 𝑓 ∈ ((2nd ‘𝑣) ↑m (1st ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
19		setcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
20	19	elexd 3474	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
21		tpex 7724	. . . 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 6978	. 2 ⊢ (𝜑 → (SetCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
24	1, 23	eqtrid 2777	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {ctp 4595  ⟨cop 4597   × cxp 5638   ∘ ccom 5644  ‘cfv 6513  (class class class)co 7389   ∈ cmpo 7391  1^st c1st 7968  2^nd c2nd 7969   ↑_m cmap 8801  ndxcnx 17169  Basecbs 17185  Hom chom 17237  compcco 17238  SetCatcsetc 18043

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-iota 6466  df-fun 6515  df-fv 6521  df-oprab 7393  df-mpo 7394  df-setc 18044

This theorem is referenced by:  setcbas  18046  setchomfval  18047  setccofval  18050