Theorem oppcval 17614

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 3457	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3		id 22	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
4		fveq2 6817	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
5		oppcval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
6	4, 5	eqtr4di 2784	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
7	6	tposeqd 8154	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (Hom ‘𝑐) = tpos 𝐻)
8	7	opeq2d 4827	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩ = ⟨(Hom ‘ndx), tpos 𝐻⟩)
9	3, 8	oveq12d 7359	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) = (𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))
10		fveq2 6817	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
11		oppcval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
12	10, 11	eqtr4di 2784	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
13	12	sqxpeqd 5643	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
14		fveq2 6817	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
15		oppcval.x	. . . . . . . . . 10 ⊢ · = (comp‘𝐶)
16	14, 15	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = · )
17	16	oveqd 7358	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)) = (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))
18	17	tposeqd 8154	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (⟨𝑧, (2nd ‘𝑢)⟩(comp‘𝑐)(1st ‘𝑢)) = tpos (⟨𝑧, (2nd ‘𝑢)⟩ · (1st ‘𝑢)))
19	13, 12, 18	mpoeq123dv 7416	. . . . . 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 4827	. . . . 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 7359	. . . 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 17613	. . . 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 7374	. . . 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 6924	. . 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	eqtrid 2778	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 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4577   × cxp 5609  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  1^st c1st 7914  2^nd c2nd 7915  tpos ctpos 8150   sSet csts 17069  ndxcnx 17099  Basecbs 17115  Hom chom 17167  compcco 17168  oppCatcoppc 17612

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-res 5623  df-iota 6432  df-fun 6478  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-tpos 8151  df-oppc 17613

This theorem is referenced by:  oppchomfval  17615  oppccofval  17617  oppcbas  17619  catcoppccl  18019