MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oppcinv Structured version   Visualization version   GIF version

Theorem oppcinv 17028
Description: An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
oppcsect.b 𝐵 = (Base‘𝐶)
oppcsect.o 𝑂 = (oppCat‘𝐶)
oppcsect.c (𝜑𝐶 ∈ Cat)
oppcsect.x (𝜑𝑋𝐵)
oppcsect.y (𝜑𝑌𝐵)
oppcinv.s 𝐼 = (Inv‘𝐶)
oppcinv.t 𝐽 = (Inv‘𝑂)
Assertion
Ref Expression
oppcinv (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))

Proof of Theorem oppcinv
StepHypRef Expression
1 incom 4153 . . 3 ((𝑋(Sect‘𝑂)𝑌) ∩ (𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌))
2 oppcsect.b . . . . . . 7 𝐵 = (Base‘𝐶)
3 oppcsect.o . . . . . . 7 𝑂 = (oppCat‘𝐶)
4 oppcsect.c . . . . . . 7 (𝜑𝐶 ∈ Cat)
5 oppcsect.y . . . . . . 7 (𝜑𝑌𝐵)
6 oppcsect.x . . . . . . 7 (𝜑𝑋𝐵)
7 eqid 2821 . . . . . . 7 (Sect‘𝐶) = (Sect‘𝐶)
8 eqid 2821 . . . . . . 7 (Sect‘𝑂) = (Sect‘𝑂)
92, 3, 4, 5, 6, 7, 8oppcsect2 17027 . . . . . 6 (𝜑 → (𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
109cnveqd 5719 . . . . 5 (𝜑(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
11 eqid 2821 . . . . . . . 8 (Hom ‘𝐶) = (Hom ‘𝐶)
12 eqid 2821 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
13 eqid 2821 . . . . . . . 8 (Id‘𝐶) = (Id‘𝐶)
142, 11, 12, 13, 7, 4, 5, 6sectss 17000 . . . . . . 7 (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
15 relxp 5546 . . . . . . 7 Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
16 relss 5629 . . . . . . 7 ((𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌(Sect‘𝐶)𝑋)))
1714, 15, 16mpisyl 21 . . . . . 6 (𝜑 → Rel (𝑌(Sect‘𝐶)𝑋))
18 dfrel2 6019 . . . . . 6 (Rel (𝑌(Sect‘𝐶)𝑋) ↔ (𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
1917, 18sylib 221 . . . . 5 (𝜑(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
2010, 19eqtrd 2856 . . . 4 (𝜑(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
212, 3, 4, 6, 5, 7, 8oppcsect2 17027 . . . 4 (𝜑 → (𝑋(Sect‘𝑂)𝑌) = (𝑋(Sect‘𝐶)𝑌))
2220, 21ineq12d 4165 . . 3 (𝜑 → ((𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ (𝑋(Sect‘𝐶)𝑌)))
231, 22syl5eq 2868 . 2 (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ (𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ (𝑋(Sect‘𝐶)𝑌)))
243, 2oppcbas 16966 . . 3 𝐵 = (Base‘𝑂)
25 oppcinv.t . . 3 𝐽 = (Inv‘𝑂)
263oppccat 16970 . . . 4 (𝐶 ∈ Cat → 𝑂 ∈ Cat)
274, 26syl 17 . . 3 (𝜑𝑂 ∈ Cat)
2824, 25, 27, 6, 5, 8invfval 17007 . 2 (𝜑 → (𝑋𝐽𝑌) = ((𝑋(Sect‘𝑂)𝑌) ∩ (𝑌(Sect‘𝑂)𝑋)))
29 oppcinv.s . . 3 𝐼 = (Inv‘𝐶)
302, 29, 4, 5, 6, 7invfval 17007 . 2 (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ (𝑋(Sect‘𝐶)𝑌)))
3123, 28, 303eqtr4d 2866 1 (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  cin 3909  wss 3910   × cxp 5526  ccnv 5527  Rel wrel 5533  cfv 6328  (class class class)co 7130  Basecbs 16461  Hom chom 16554  compcco 16555  Catccat 16913  Idccid 16914  oppCatcoppc 16959  Sectcsect 16992  Invcinv 16993
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-tpos 7867  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-dec 12077  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-hom 16567  df-cco 16568  df-cat 16917  df-cid 16918  df-oppc 16960  df-sect 16995  df-inv 16996
This theorem is referenced by:  oppciso  17029  episect  17033
  Copyright terms: Public domain W3C validator