Theorem oppccomfpropd 16999

Description: If two categories have the same hom-sets and composition, so do their opposites. (Contributed by Mario Carneiro, 26-Jan-2017.)

Hypotheses

Ref	Expression
oppchomfpropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
oppccomfpropd.1	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))

Assertion

Ref	Expression
oppccomfpropd	⊢ (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷)))

Proof of Theorem oppccomfpropd

Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2823	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2823	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2823	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2823	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		oppchomfpropd.1	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → (Homf ‘𝐶) = (Homf ‘𝐷))
7		oppccomfpropd.1	. . . . . . 7 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
8	7	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → (compf‘𝐶) = (compf‘𝐷))
9		simplr3 1213	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑧 ∈ (Base‘𝐶))
10		simplr2 1212	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑦 ∈ (Base‘𝐶))
11		simplr1 1211	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑥 ∈ (Base‘𝐶))
12		simprr 771	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))
13		eqid 2823	. . . . . . . 8 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
14	2, 13	oppchom 16987	. . . . . . 7 ⊢ (𝑦(Hom ‘(oppCat‘𝐶))𝑧) = (𝑧(Hom ‘𝐶)𝑦)
15	12, 14	eleqtrdi 2925	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑔 ∈ (𝑧(Hom ‘𝐶)𝑦))
16		simprl 769	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦))
17	2, 13	oppchom 16987	. . . . . . 7 ⊢ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) = (𝑦(Hom ‘𝐶)𝑥)
18	16, 17	eleqtrdi 2925	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑓 ∈ (𝑦(Hom ‘𝐶)𝑥))
19	1, 2, 3, 4, 6, 8, 9, 10, 11, 15, 18	comfeqval 16980	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → (𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔) = (𝑓(⟨𝑧, 𝑦⟩(comp‘𝐷)𝑥)𝑔))
20	1, 3, 13, 11, 10, 9	oppcco 16989	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐶))𝑧)𝑓) = (𝑓(⟨𝑧, 𝑦⟩(comp‘𝐶)𝑥)𝑔))
21		eqid 2823	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
22		eqid 2823	. . . . . 6 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
23	5	homfeqbas 16968	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
24	23	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → (Base‘𝐶) = (Base‘𝐷))
25	11, 24	eleqtrd 2917	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑥 ∈ (Base‘𝐷))
26	10, 24	eleqtrd 2917	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑦 ∈ (Base‘𝐷))
27	9, 24	eleqtrd 2917	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → 𝑧 ∈ (Base‘𝐷))
28	21, 4, 22, 25, 26, 27	oppcco 16989	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐷))𝑧)𝑓) = (𝑓(⟨𝑧, 𝑦⟩(comp‘𝐷)𝑥)𝑔))
29	19, 20, 28	3eqtr4d 2868	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) ∧ (𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐶))𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐷))𝑧)𝑓))
30	29	ralrimivva 3193	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦)∀𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐶))𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐷))𝑧)𝑓))
31	30	ralrimivvva 3194	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦)∀𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐶))𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐷))𝑧)𝑓))
32		eqid 2823	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33		eqid 2823	. . 3 ⊢ (comp‘(oppCat‘𝐷)) = (comp‘(oppCat‘𝐷))
34		eqid 2823	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
35	13, 1	oppcbas 16990	. . . 4 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
36	35	a1i 11	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐶)))
37	22, 21	oppcbas 16990	. . . 4 ⊢ (Base‘𝐷) = (Base‘(oppCat‘𝐷))
38	23, 37	syl6eq 2874	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐷)))
39	5	oppchomfpropd 16998	. . 3 ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷)))
40	32, 33, 34, 36, 38, 39	comfeq 16978	. 2 ⊢ (𝜑 → ((compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷)) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘(oppCat‘𝐶))𝑦)∀𝑔 ∈ (𝑦(Hom ‘(oppCat‘𝐶))𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐶))𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝐷))𝑧)𝑓)))
41	31, 40	mpbird 259	1 ⊢ (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ⟨cop 4575  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  Hom chom 16578  compcco 16579  Hom_f chomf 16939  comp_fccomf 16940  oppCatcoppc 16983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-tpos 7894  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-hom 16591  df-cco 16592  df-homf 16943  df-comf 16944  df-oppc 16984

This theorem is referenced by:  yonpropd  17520