Theorem oppcval 16373

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 3212	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3		id 22	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
4		fveq2 6191	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
5		oppcval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
6	4, 5	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
7	6	tposeqd 7355	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (Hom ‘𝑐) = tpos 𝐻)
8	7	opeq2d 4409	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩ = ⟨(Hom ‘ndx), tpos 𝐻⟩)
9	3, 8	oveq12d 6668	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) = (𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))
10		fveq2 6191	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
11		oppcval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
12	10, 11	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
13	12	sqxpeqd 5141	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
14		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
15		oppcval.x	. . . . . . . . . 10 ⊢ · = (comp‘𝐶)
16	14, 15	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = · )
17	16	oveqd 6667	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)) = (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))
18	17	tposeqd 7355	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)) = tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))
19	13, 12, 18	mpt2eq123dv 6717	. . . . . 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 4409	. . . . 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 6668	. . . 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 16372	. . . 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 6678	. . . 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 6282	. . 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 17	. 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	syl5eq 2668	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 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   × cxp 5112  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167  tpos ctpos 7351  ndxcnx 15854   sSet csts 15855  Basecbs 15857  Hom chom 15952  compcco 15953  oppCatcoppc 16371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-tpos 7352  df-oppc 16372

This theorem is referenced by:  oppchomfval  16374  oppccofval  16376  oppcbas  16378  catcoppccl  16758