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Theorem oppcval 17585
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 3461 . . 3 (𝐢 ∈ 𝑉 β†’ 𝐢 ∈ V)
3 id 22 . . . . . 6 (𝑐 = 𝐢 β†’ 𝑐 = 𝐢)
4 fveq2 6839 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = (Hom β€˜πΆ))
5 oppcval.h . . . . . . . . 9 𝐻 = (Hom β€˜πΆ)
64, 5eqtr4di 2794 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Hom β€˜π‘) = 𝐻)
76tposeqd 8156 . . . . . . 7 (𝑐 = 𝐢 β†’ tpos (Hom β€˜π‘) = tpos 𝐻)
87opeq2d 4835 . . . . . 6 (𝑐 = 𝐢 β†’ ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘)⟩ = ⟨(Hom β€˜ndx), tpos 𝐻⟩)
93, 8oveq12d 7371 . . . . 5 (𝑐 = 𝐢 β†’ (𝑐 sSet ⟨(Hom β€˜ndx), tpos (Hom β€˜π‘)⟩) = (𝐢 sSet ⟨(Hom β€˜ndx), tpos 𝐻⟩))
10 fveq2 6839 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΆ))
11 oppcval.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΆ)
1210, 11eqtr4di 2794 . . . . . . . 8 (𝑐 = 𝐢 β†’ (Baseβ€˜π‘) = 𝐡)
1312sqxpeqd 5663 . . . . . . 7 (𝑐 = 𝐢 β†’ ((Baseβ€˜π‘) Γ— (Baseβ€˜π‘)) = (𝐡 Γ— 𝐡))
14 fveq2 6839 . . . . . . . . . 10 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = (compβ€˜πΆ))
15 oppcval.x . . . . . . . . . 10 Β· = (compβ€˜πΆ)
1614, 15eqtr4di 2794 . . . . . . . . 9 (𝑐 = 𝐢 β†’ (compβ€˜π‘) = Β· )
1716oveqd 7370 . . . . . . . 8 (𝑐 = 𝐢 β†’ (βŸ¨π‘§, (2nd β€˜π‘’)⟩(compβ€˜π‘)(1st β€˜π‘’)) = (βŸ¨π‘§, (2nd β€˜π‘’)⟩ Β· (1st β€˜π‘’)))
1817tposeqd 8156 . . . . . . 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 7428 . . . . . 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 4835 . . . . 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 7371 . . . 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 17584 . . . 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 7386 . . . 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 6945 . . 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 2788 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 1541   ∈ wcel 2106  Vcvv 3443  βŸ¨cop 4590   Γ— cxp 5629  β€˜cfv 6493  (class class class)co 7353   ∈ cmpo 7355  1st c1st 7915  2nd c2nd 7916  tpos ctpos 8152   sSet csts 17027  ndxcnx 17057  Basecbs 17075  Hom chom 17136  compcco 17137  oppCatcoppc 17583
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-tpos 8153  df-oppc 17584
This theorem is referenced by:  oppchomfval  17586  oppchomfvalOLD  17587  oppccofval  17589  oppcbas  17591  oppcbasOLD  17592  catcoppccl  17995  catcoppcclOLD  17996
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