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Theorem oppcval 16574
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 (𝐶𝑉𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
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.
StepHypRef Expression
1 oppcval.o . 2 𝑂 = (oppCat‘𝐶)
2 elex 3352 . . 3 (𝐶𝑉𝐶 ∈ V)
3 id 22 . . . . . 6 (𝑐 = 𝐶𝑐 = 𝐶)
4 fveq2 6352 . . . . . . . . 9 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
5 oppcval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
64, 5syl6eqr 2812 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
76tposeqd 7524 . . . . . . 7 (𝑐 = 𝐶 → tpos (Hom ‘𝑐) = tpos 𝐻)
87opeq2d 4560 . . . . . 6 (𝑐 = 𝐶 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩ = ⟨(Hom ‘ndx), tpos 𝐻⟩)
93, 8oveq12d 6831 . . . . 5 (𝑐 = 𝐶 → (𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) = (𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))
10 fveq2 6352 . . . . . . . . 9 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
11 oppcval.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
1210, 11syl6eqr 2812 . . . . . . . 8 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
1312sqxpeqd 5298 . . . . . . 7 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
14 fveq2 6352 . . . . . . . . . 10 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
15 oppcval.x . . . . . . . . . 10 · = (comp‘𝐶)
1614, 15syl6eqr 2812 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = · )
1716oveqd 6830 . . . . . . . 8 (𝑐 = 𝐶 → (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)) = (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))
1817tposeqd 7524 . . . . . . 7 (𝑐 = 𝐶 → tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)) = tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))
1913, 12, 18mpt2eq123dv 6882 . . . . . 6 (𝑐 = 𝐶 → (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢))) = (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢))))
2019opeq2d 4560 . . . . 5 (𝑐 = 𝐶 → ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)))⟩ = ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩)
219, 20oveq12d 6831 . . . 4 (𝑐 = 𝐶 → ((𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)))⟩) = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
22 df-oppc 16573 . . . 4 oppCat = (𝑐 ∈ V ↦ ((𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ ((Base‘𝑐) × (Base‘𝑐)), 𝑧 ∈ (Base‘𝑐) ↦ tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)))⟩))
23 ovex 6841 . . . 4 ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩) ∈ V
2421, 22, 23fvmpt 6444 . . 3 (𝐶 ∈ V → (oppCat‘𝐶) = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
252, 24syl 17 . 2 (𝐶𝑉 → (oppCat‘𝐶) = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
261, 25syl5eq 2806 1 (𝐶𝑉𝑂 = ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327   × cxp 5264  cfv 6049  (class class class)co 6813  cmpt2 6815  1st c1st 7331  2nd c2nd 7332  tpos ctpos 7520  ndxcnx 16056   sSet csts 16057  Basecbs 16059  Hom chom 16154  compcco 16155  oppCatcoppc 16572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-tpos 7521  df-oppc 16573
This theorem is referenced by:  oppchomfval  16575  oppccofval  16577  oppcbas  16579  catcoppccl  16959
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