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Theorem oppcval 16580
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 3364 . . 3 (𝐶𝑉𝐶 ∈ V)
3 id 22 . . . . . 6 (𝑐 = 𝐶𝑐 = 𝐶)
4 fveq2 6332 . . . . . . . . 9 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
5 oppcval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
64, 5syl6eqr 2823 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
76tposeqd 7507 . . . . . . 7 (𝑐 = 𝐶 → tpos (Hom ‘𝑐) = tpos 𝐻)
87opeq2d 4546 . . . . . 6 (𝑐 = 𝐶 → ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩ = ⟨(Hom ‘ndx), tpos 𝐻⟩)
93, 8oveq12d 6811 . . . . 5 (𝑐 = 𝐶 → (𝑐 sSet ⟨(Hom ‘ndx), tpos (Hom ‘𝑐)⟩) = (𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩))
10 fveq2 6332 . . . . . . . . 9 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
11 oppcval.b . . . . . . . . 9 𝐵 = (Base‘𝐶)
1210, 11syl6eqr 2823 . . . . . . . 8 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
1312sqxpeqd 5281 . . . . . . 7 (𝑐 = 𝐶 → ((Base‘𝑐) × (Base‘𝑐)) = (𝐵 × 𝐵))
14 fveq2 6332 . . . . . . . . . 10 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
15 oppcval.x . . . . . . . . . 10 · = (comp‘𝐶)
1614, 15syl6eqr 2823 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = · )
1716oveqd 6810 . . . . . . . 8 (𝑐 = 𝐶 → (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)) = (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))
1817tposeqd 7507 . . . . . . 7 (𝑐 = 𝐶 → tpos (⟨𝑧, (2nd𝑢)⟩(comp‘𝑐)(1st𝑢)) = tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))
1913, 12, 18mpt2eq123dv 6864 . . . . . 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 4546 . . . . 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 6811 . . . 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 16579 . . . 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 6823 . . . 4 ((𝐶 sSet ⟨(Hom ‘ndx), tpos 𝐻⟩) sSet ⟨(comp‘ndx), (𝑢 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ tpos (⟨𝑧, (2nd𝑢)⟩ · (1st𝑢)))⟩) ∈ V
2421, 22, 23fvmpt 6424 . . 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 2817 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 1631  wcel 2145  Vcvv 3351  cop 4322   × cxp 5247  cfv 6031  (class class class)co 6793  cmpt2 6795  1st c1st 7313  2nd c2nd 7314  tpos ctpos 7503  ndxcnx 16061   sSet csts 16062  Basecbs 16064  Hom chom 16160  compcco 16161  oppCatcoppc 16578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-tpos 7504  df-oppc 16579
This theorem is referenced by:  oppchomfval  16581  oppccofval  16583  oppcbas  16585  catcoppccl  16965
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