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Theorem oppcval 17041
 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 3428 . . 3 (𝐶𝑉𝐶 ∈ V)
3 id 22 . . . . . 6 (𝑐 = 𝐶𝑐 = 𝐶)
4 fveq2 6658 . . . . . . . . 9 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
5 oppcval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
64, 5eqtr4di 2811 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
76tposeqd 7905 . . . . . . 7 (𝑐 = 𝐶 → tpos (Hom ‘𝑐) = tpos 𝐻)
87opeq2d 4770 . . . . . 6 (𝑐 = 𝐶 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩ = ⟨(Hom ‘ndx), tpos 𝐻⟩)
93, 8oveq12d 7168 . . . . 5 (𝑐 = 𝐶 → (𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) = (𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))
10 fveq2 6658 . . . . . . . . 9 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
11 oppcval.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
1210, 11eqtr4di 2811 . . . . . . . 8 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
1312sqxpeqd 5556 . . . . . . 7 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
14 fveq2 6658 . . . . . . . . . 10 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
15 oppcval.x . . . . . . . . . 10 · = (comp‘𝐶)
1614, 15eqtr4di 2811 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = · )
1716oveqd 7167 . . . . . . . 8 (𝑐 = 𝐶 → (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)) = (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))
1817tposeqd 7905 . . . . . . 7 (𝑐 = 𝐶 → tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)) = tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))
1913, 12, 18mpoeq123dv 7223 . . . . . 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 4770 . . . . 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 7168 . . . 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 17040 . . . 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 7183 . . . 4 ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩) ∈ V
2421, 22, 23fvmpt 6759 . . 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 2805 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 1538   ∈ wcel 2111  Vcvv 3409  ⟨cop 4528   × cxp 5522  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152  1st c1st 7691  2nd c2nd 7692  tpos ctpos 7901  ndxcnx 16538   sSet csts 16539  Basecbs 16541  Hom chom 16634  compcco 16635  oppCatcoppc 17039 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-res 5536  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-tpos 7902  df-oppc 17040 This theorem is referenced by:  oppchomfval  17042  oppccofval  17044  oppcbas  17046  catcoppccl  17434
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