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Theorem 2oppccomf 16591
Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 16603. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypothesis
Ref Expression
oppcbas.1 𝑂 = (oppCat‘𝐶)
Assertion
Ref Expression
2oppccomf (compf𝐶) = (compf‘(oppCat‘𝑂))

Proof of Theorem 2oppccomf
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppcbas.1 . . . . . . . . 9 𝑂 = (oppCat‘𝐶)
2 eqid 2771 . . . . . . . . 9 (Base‘𝐶) = (Base‘𝐶)
31, 2oppcbas 16584 . . . . . . . 8 (Base‘𝐶) = (Base‘𝑂)
4 eqid 2771 . . . . . . . 8 (comp‘𝑂) = (comp‘𝑂)
5 eqid 2771 . . . . . . . 8 (oppCat‘𝑂) = (oppCat‘𝑂)
6 simpr1 1233 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
7 simpr2 1235 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
8 simpr3 1237 . . . . . . . 8 ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
93, 4, 5, 6, 7, 8oppcco 16583 . . . . . . 7 ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓) = (𝑓(⟨𝑧, 𝑦⟩(comp‘𝑂)𝑥)𝑔))
10 eqid 2771 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
112, 10, 1, 8, 7, 6oppcco 16583 . . . . . . 7 ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑓(⟨𝑧, 𝑦⟩(comp‘𝑂)𝑥)𝑔) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
129, 11eqtr2d 2806 . . . . . 6 ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓))
1312ralrimivw 3116 . . . . 5 ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓))
1413ralrimivw 3116 . . . 4 ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓))
1514ralrimivvva 3121 . . 3 (⊤ → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓))
16 eqid 2771 . . . 4 (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂))
17 eqid 2771 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
18 eqidd 2772 . . . 4 (⊤ → (Base‘𝐶) = (Base‘𝐶))
191, 22oppcbas 16589 . . . . 5 (Base‘𝐶) = (Base‘(oppCat‘𝑂))
2019a1i 11 . . . 4 (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂)))
2112oppchomf 16590 . . . . 5 (Homf𝐶) = (Homf ‘(oppCat‘𝑂))
2221a1i 11 . . . 4 (⊤ → (Homf𝐶) = (Homf ‘(oppCat‘𝑂)))
2310, 16, 17, 18, 20, 22comfeq 16572 . . 3 (⊤ → ((compf𝐶) = (compf‘(oppCat‘𝑂)) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓)))
2415, 23mpbird 247 . 2 (⊤ → (compf𝐶) = (compf‘(oppCat‘𝑂)))
2524trud 1641 1 (compf𝐶) = (compf‘(oppCat‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wa 382  w3a 1071   = wceq 1631  wtru 1632  wcel 2145  wral 3061  cop 4323  cfv 6030  (class class class)co 6795  Basecbs 16063  Hom chom 16159  compcco 16160  Homf chomf 16533  compfccomf 16534  oppCatcoppc 16577
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-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  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-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-tpos 7507  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-er 7899  df-en 8113  df-dom 8114  df-sdom 8115  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-nn 11226  df-2 11284  df-3 11285  df-4 11286  df-5 11287  df-6 11288  df-7 11289  df-8 11290  df-9 11291  df-n0 11499  df-z 11584  df-dec 11700  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-hom 16173  df-cco 16174  df-homf 16537  df-comf 16538  df-oppc 16578
This theorem is referenced by:  oppcepi  16605  oppchofcl  17107  oppcyon  17116  oyoncl  17117
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