Theorem oppcval 17765

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 3484	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3		id 23	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
4		fveq2 6879	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
5		oppcval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
6	4, 5	eqtr4di 2822	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
7	6	tposeqd 8221	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (Hom ‘𝑐) = tpos 𝐻)
8	7	opeq2d 4846	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩ = ⟨(Hom ‘ndx), tpos 𝐻⟩)
9	3, 8	oveq12d 7426	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) = (𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))
10		fveq2 6879	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
11		oppcval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
12	10, 11	eqtr4di 2822	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
13	12	sqxpeqd 5691	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
14		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
15		oppcval.x	. . . . . . . . . 10 ⊢ · = (comp‘𝐶)
16	14, 15	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = · )
17	16	oveqd 7425	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)) = (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))
18	17	tposeqd 8221	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)) = tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))
19	13, 12, 18	mpoeq123dv 7483	. . . . . 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 4846	. . . . 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 7426	. . . 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 17764	. . . 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 7441	. . . 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 6987	. . 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 2816	1 ⊢ (𝐶 ∈ 𝑉 → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))⟩))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4597   × cxp 5657  ‘cfv 6533  (class class class)co 7408   ∈ cmpo 7410  1^st c1st 7980  2^nd c2nd 7981  tpos ctpos 8217   sSet csts 17219  ndxcnx 17249  Basecbs 17265  Hom chom 17317  compcco 17318  oppCatcoppc 17763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-iota 6489  df-fun 6535  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-tpos 8218  df-oppc 17764

This theorem is referenced by:  oppchomfval  17766  oppccofval  17768  oppcbas  17770  catcoppccl  18170