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Theorem oppcval 17700
Description: Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
oppcval.b 𝐡 = (Baseβ€˜πΆ)
oppcval.h 𝐻 = (Hom β€˜πΆ)
oppcval.x Β· = (compβ€˜πΆ)
oppcval.o 𝑂 = (oppCatβ€˜πΆ)
Assertion
Ref Expression
oppcval (𝐢 ∈ 𝑉 β†’ 𝑂 = ((𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩))
Distinct variable group:   𝑧,𝑒,𝐢
Allowed substitution hints:   𝐡(𝑧,𝑒)   Β· (𝑧,𝑒)   𝐻(𝑧,𝑒)   𝑂(𝑧,𝑒)   𝑉(𝑧,𝑒)

Proof of Theorem oppcval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 oppcval.o . 2 𝑂 = (oppCatβ€˜πΆ)
2 elex 3492 . . 3 (𝐢 ∈ 𝑉 β†’ 𝐢 ∈ V)
3 id 22 . . . . . 6 (𝑐 = 𝐢 β†’ 𝑐 = 𝐢)
4 fveq2 6902 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
5 oppcval.h . . . . . . . . 9 𝐻 = (Hom β€˜πΆ)
64, 5eqtr4di 2786 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = 𝐻)
76tposeqd 8241 . . . . . . 7 (𝑐 = 𝐢 β†’ tpos (Hom β€˜π‘) = tpos 𝐻)
87opeq2d 4885 . . . . . 6 (𝑐 = 𝐢 β†’ ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘)⟩ = ⟨(Hom β€˜ndx), tpos 𝐻⟩)
93, 8oveq12d 7444 . . . . 5 (𝑐 = 𝐢 β†’ (𝑐 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘)⟩) = (𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩))
10 fveq2 6902 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
11 oppcval.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΆ)
1210, 11eqtr4di 2786 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
1312sqxpeqd 5714 . . . . . . 7 (𝑐 = 𝐢 β†’ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)) = (𝐡 Γ— 𝐡))
14 fveq2 6902 . . . . . . . . . 10 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = (compβ€˜πΆ))
15 oppcval.x . . . . . . . . . 10 Β· = (compβ€˜πΆ)
1614, 15eqtr4di 2786 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = Β· )
1716oveqd 7443 . . . . . . . 8 (𝑐 = 𝐢 β†’ (βŸ¨π‘§, (2nd β€˜π‘’)⟩(compβ€˜π‘)(1st β€˜π‘’)) = (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))
1817tposeqd 8241 . . . . . . 7 (𝑐 = 𝐢 β†’ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩(compβ€˜π‘)(1st β€˜π‘’)) = tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))
1913, 12, 18mpoeq123dv 7501 . . . . . 6 (𝑐 = 𝐢 β†’ (𝑒 ∈ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)), 𝑧 ∈ (Baseβ€˜π‘) ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩(compβ€˜π‘)(1st β€˜π‘’))) = (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’))))
2019opeq2d 4885 . . . . 5 (𝑐 = 𝐢 β†’ ⟨(compβ€˜ndx), (𝑒 ∈ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)), 𝑧 ∈ (Baseβ€˜π‘) ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩(compβ€˜π‘)(1st β€˜π‘’)))⟩ = ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩)
219, 20oveq12d 7444 . . . 4 (𝑐 = 𝐢 β†’ ((𝑐 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘)⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)), 𝑧 ∈ (Baseβ€˜π‘) ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩(compβ€˜π‘)(1st β€˜π‘’)))⟩) = ((𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩))
22 df-oppc 17699 . . . 4 oppCat = (𝑐 ∈ V ↦ ((𝑐 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘)⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)), 𝑧 ∈ (Baseβ€˜π‘) ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩(compβ€˜π‘)(1st β€˜π‘’)))⟩))
23 ovex 7459 . . . 4 ((𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩) ∈ V
2421, 22, 23fvmpt 7010 . . 3 (𝐢 ∈ V β†’ (oppCatβ€˜πΆ) = ((𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩))
252, 24syl 17 . 2 (𝐢 ∈ 𝑉 β†’ (oppCatβ€˜πΆ) = ((𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩))
261, 25eqtrid 2780 1 (𝐢 ∈ 𝑉 β†’ 𝑂 = ((𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩) sSet ⟨(compβ€˜ndx), (𝑒 ∈ (𝐡 Γ— 𝐡), 𝑧 ∈ 𝐡 ↦ tpos (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))⟩))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3473  βŸ¨cop 4638   Γ— cxp 5680  β€˜cfv 6553  (class class class)co 7426   ∈ cmpo 7428  1st c1st 7997  2nd c2nd 7998  tpos ctpos 8237   sSet csts 17139  ndxcnx 17169  Basecbs 17187  Hom chom 17251  compcco 17252  oppCatcoppc 17698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-tpos 8238  df-oppc 17699
This theorem is referenced by:  oppchomfval  17701  oppchomfvalOLD  17702  oppccofval  17704  oppcbas  17706  oppcbasOLD  17707  catcoppccl  18113  catcoppcclOLD  18114
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