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Theorem oppcval 17653

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, (2^nd ‘𝑢)⟩ · (1^st ‘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 3492	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3		id 22	. . . . . 6 ⊢ (𝑐 = 𝐶 → 𝑐 = 𝐶)
4		fveq2 6888	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
5		oppcval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
6	4, 5	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
7	6	tposeqd 8210	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (Hom ‘𝑐) = tpos 𝐻)
8	7	opeq2d 4879	. . . . . 6 ⊢ (𝑐 = 𝐶 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩ = ⟨(Hom ‘ndx), tpos 𝐻⟩)
9	3, 8	oveq12d 7423	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) = (𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))
10		fveq2 6888	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
11		oppcval.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐶)
12	10, 11	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
13	12	sqxpeqd 5707	. . . . . . 7 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
14		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
15		oppcval.x	. . . . . . . . . 10 ⊢ · = (comp‘𝐶)
16	14, 15	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑐 = 𝐶 → (comp‘𝑐) = · )
17	16	oveqd 7422	. . . . . . . 8 ⊢ (𝑐 = 𝐶 → (⟨𝑧, (2^nd ‘𝑢)⟩(comp‘𝑐)(1^st ‘𝑢)) = (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))
18	17	tposeqd 8210	. . . . . . 7 ⊢ (𝑐 = 𝐶 → tpos (⟨𝑧, (2^nd ‘𝑢)⟩(comp‘𝑐)(1^st ‘𝑢)) = tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))
19	13, 12, 18	mpoeq123dv 7480	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩(comp‘𝑐)(1^st ‘𝑢))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢))))
20	19	opeq2d 4879	. . . . 5 ⊢ (𝑐 = 𝐶 → ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩(comp‘𝑐)(1^st ‘𝑢)))⟩ = ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩)
21	9, 20	oveq12d 7423	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩(comp‘𝑐)(1^st ‘𝑢)))⟩) = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩))
22		df-oppc 17652	. . . 4 ⊢ oppCat = (𝑐 ∈ V ↦ ((𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩(comp‘𝑐)(1^st ‘𝑢)))⟩))
23		ovex 7438	. . . 4 ⊢ ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩) ∈ V
24	21, 22, 23	fvmpt 6995	. . 3 ⊢ (𝐶 ∈ V → (oppCat‘𝐶) = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘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, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩))
26	1, 25	eqtrid 2784	1 ⊢ (𝐶 ∈ 𝑉 → 𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ tpos (⟨𝑧, (2^nd ‘𝑢)⟩ · (1^st ‘𝑢)))⟩))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟨cop 4633 × cxp 5673 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 1^st c1st 7969 2^nd c2nd 7970 tpos ctpos 8206 sSet csts 17092 ndxcnx 17122 Basecbs 17140 Hom chom 17204 compcco 17205 oppCatcoppc 17651

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-res 5687 df-iota 6492 df-fun 6542 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-tpos 8207 df-oppc 17652

This theorem is referenced by: oppchomfval 17654 oppchomfvalOLD 17655 oppccofval 17657 oppcbas 17659 oppcbasOLD 17660 catcoppccl 18063 catcoppcclOLD 18064

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