Theorem oppccomfpropd 17744

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 2736	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2736	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2736	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2736	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		oppchomfpropd.1	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	ad2antrr 726	. . . . . 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 726	. . . . . 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 1218	. . . . . 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 1217	. . . . . 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 1216	. . . . . 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 772	. . . . . . 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 2736	. . . . . . . 8 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
14	2, 13	oppchom 17732	. . . . . . 7 ⊢ (𝑦(Hom ‘(oppCat‘𝐶))𝑧) = (𝑧(Hom ‘𝐶)𝑦)
15	12, 14	eleqtrdi 2845	. . . . . 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 770	. . . . . . 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 17732	. . . . . . 7 ⊢ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) = (𝑦(Hom ‘𝐶)𝑥)
18	16, 17	eleqtrdi 2845	. . . . . 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 17725	. . . . 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 17734	. . . . 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 2736	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
22		eqid 2736	. . . . . 6 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
23	5	homfeqbas 17713	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
24	23	ad2antrr 726	. . . . . . 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 2837	. . . . . 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 2837	. . . . . 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 2837	. . . . . 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 17734	. . . . 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 2781	. . . 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 3188	. . 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 3191	. 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 2736	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33		eqid 2736	. . 3 ⊢ (comp‘(oppCat‘𝐷)) = (comp‘(oppCat‘𝐷))
34		eqid 2736	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
35	13, 1	oppcbas 17735	. . . 4 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
36	35	a1i 11	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐶)))
37	22, 21	oppcbas 17735	. . . 4 ⊢ (Base‘𝐷) = (Base‘(oppCat‘𝐷))
38	23, 37	eqtrdi 2787	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐷)))
39	5	oppchomfpropd 17743	. . 3 ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷)))
40	32, 33, 34, 36, 38, 39	comfeq 17723	. 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 257	1 ⊢ (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ⟨cop 4612  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  Hom chom 17287  compcco 17288  Hom_f chomf 17683  comp_fccomf 17684  oppCatcoppc 17728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-hom 17300  df-cco 17301  df-homf 17687  df-comf 17688  df-oppc 17729

This theorem is referenced by:  yonpropd  18285