Theorem oppcval 17937

Description: Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
oppcval.b	⊢ 𝐵 = (Base‘𝐶)
oppcval.h	⊢ 𝐻 = (Hom ‘𝐶)
oppcval.x	⊢ · = (comp‘𝐶)
oppcval.o	⊢ 𝑂 = (oppCat‘𝐶)

Assertion

Ref	Expression
oppcval	⊢ (𝐶 ∈ 𝑉 → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))

Distinct variable group:   𝑧,𝑢,𝐶

Allowed substitution hints:   𝐵(𝑧, 𝑢)   · (𝑧, 𝑢)   𝐻(𝑧, 𝑢)   𝑂(𝑧, 𝑢)   𝑉(𝑧, 𝑢)

Proof of Theorem oppcval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppcval.o	. 2 ⊢ 𝑂 = (oppCat‘𝐶)
2		elex 3475	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3		id 23	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
4		fveq2 6916	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
5		oppcval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
6	4, 5	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
7	6	tposeqd 8284	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (Hom ‘𝑐) = tpos 𝐻)
8	7	opeq2d 4852	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩ = ⟨(Hom ‘ndx), tpos 𝐻⟩)
9	3, 8	oveq12d 7469	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) = (𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))
10		fveq2 6916	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
11		oppcval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
12	10, 11	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
13	12	sqxpeqd 5690	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
14		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
15		oppcval.x	. . . . . . . . . 10 ⊢ · = (comp‘𝐶)
16	14, 15	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = · )
17	16	oveqd 7468	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)) = (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))
18	17	tposeqd 8284	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)) = tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))
19	13, 12, 18	mpoeq123dv 7526	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢))))
20	19	opeq2d 4852	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)))⟩ = ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩)
21	9, 20	oveq12d 7469	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)))⟩) = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))
22		df-oppc 17936	. . . 4 ⊢ oppCat = (𝑐 ∈ V ↦ ((𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)))⟩))
23		ovex 7484	. . . 4 ⊢ ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩) ∈ V
24	21, 22, 23	fvmpt 7024	. . 3 ⊢ (𝐶 ∈ V → (oppCat‘𝐶) = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))
25	2, 24	syl 18	. 2 ⊢ (𝐶 ∈ 𝑉 → (oppCat‘𝐶) = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))
26	1, 25	eqtrid 2810	1 ⊢ (𝐶 ∈ 𝑉 → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840   × cxp 5656  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  1^st c1st 8033  2^nd c2nd 8034  tpos ctpos 8280   sSet csts 17385  ndxcnx 17415  Basecbs 17431  Hom chom 17483  compcco 17484  oppCatcoppc 17935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-res 5670  df-iota 6520  df-fun 6568  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-tpos 8281  df-oppc 17936

This theorem is used by:  oppchomfval  17938  oppccofval  17940  oppcbas  17942  catcoppccl  18342