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Theorem oppcval 17663
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 3487 . . 3 (𝐢 ∈ 𝑉 β†’ 𝐢 ∈ V)
3 id 22 . . . . . 6 (𝑐 = 𝐢 β†’ 𝑐 = 𝐢)
4 fveq2 6884 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
5 oppcval.h . . . . . . . . 9 𝐻 = (Hom β€˜πΆ)
64, 5eqtr4di 2784 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = 𝐻)
76tposeqd 8212 . . . . . . 7 (𝑐 = 𝐢 β†’ tpos (Hom β€˜π‘) = tpos 𝐻)
87opeq2d 4875 . . . . . 6 (𝑐 = 𝐢 β†’ ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘)⟩ = ⟨(Hom β€˜ndx), tpos 𝐻⟩)
93, 8oveq12d 7422 . . . . 5 (𝑐 = 𝐢 β†’ (𝑐 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘)⟩) = (𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩))
10 fveq2 6884 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
11 oppcval.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΆ)
1210, 11eqtr4di 2784 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
1312sqxpeqd 5701 . . . . . . 7 (𝑐 = 𝐢 β†’ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)) = (𝐡 Γ— 𝐡))
14 fveq2 6884 . . . . . . . . . 10 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = (compβ€˜πΆ))
15 oppcval.x . . . . . . . . . 10 Β· = (compβ€˜πΆ)
1614, 15eqtr4di 2784 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = Β· )
1716oveqd 7421 . . . . . . . 8 (𝑐 = 𝐢 β†’ (βŸ¨π‘§, (2nd β€˜π‘’)⟩(compβ€˜π‘)(1st β€˜π‘’)) = (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))
1817tposeqd 8212 . . . . . . 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 7479 . . . . . 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 4875 . . . . 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 7422 . . . 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 17662 . . . 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 7437 . . . 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 6991 . . 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 2778 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 3468  βŸ¨cop 4629   Γ— cxp 5667  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  1st c1st 7969  2nd c2nd 7970  tpos ctpos 8208   sSet csts 17102  ndxcnx 17132  Basecbs 17150  Hom chom 17214  compcco 17215  oppCatcoppc 17661
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-tpos 8209  df-oppc 17662
This theorem is referenced by:  oppchomfval  17664  oppchomfvalOLD  17665  oppccofval  17667  oppcbas  17669  oppcbasOLD  17670  catcoppccl  18076  catcoppcclOLD  18077
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