Theorem oppccomfpropd 17908

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 2760	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2760	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2760	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		oppchomfpropd.1	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	ad2antrr 739	. . . . . 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 739	. . . . . 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 1236	. . . . . 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 1235	. . . . . 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 1234	. . . . . 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 785	. . . . . . 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 2760	. . . . . . . 8 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
14	2, 13	oppchom 17896	. . . . . . 7 ⊢ (𝑦(Hom ‘(oppCat‘𝐶))𝑧) = (𝑧(Hom ‘𝐶)𝑦)
15	12, 14	eleqtrdi 2870	. . . . . 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 783	. . . . . . 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 17896	. . . . . . 7 ⊢ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) = (𝑦(Hom ‘𝐶)𝑥)
18	16, 17	eleqtrdi 2870	. . . . . 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 17889	. . . . 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 17898	. . . . 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 2760	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
22		eqid 2760	. . . . . 6 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
23	5	homfeqbas 17877	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
24	23	ad2antrr 739	. . . . . . 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 2862	. . . . . 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 2862	. . . . . 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 2862	. . . . . 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 17898	. . . . 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 2805	. . . 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 3205	. . 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 3208	. 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 2760	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33		eqid 2760	. . 3 ⊢ (comp‘(oppCat‘𝐷)) = (comp‘(oppCat‘𝐷))
34		eqid 2760	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
35	13, 1	oppcbas 17899	. . . 4 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
36	35	a1i 11	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐶)))
37	22, 21	oppcbas 17899	. . . 4 ⊢ (Base‘𝐷) = (Base‘(oppCat‘𝐷))
38	23, 37	eqtrdi 2811	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐷)))
39	5	oppchomfpropd 17907	. . 3 ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷)))
40	32, 33, 34, 36, 38, 39	comfeq 17887	. 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 260	1 ⊢ (𝜑 → (compf‘(oppCat‘𝐶)) = (compf‘(oppCat‘𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ⟨cop 4834  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  Hom chom 17446  compcco 17447  Hom_f chomf 17847  comp_fccomf 17848  oppCatcoppc 17892

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-tpos 8248  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-er 8722  df-en 8979  df-dom 8980  df-sdom 8981  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-3 12411  df-4 12412  df-5 12413  df-6 12414  df-7 12415  df-8 12416  df-9 12417  df-n0 12612  df-z 12699  df-dec 12820  df-sets 17349  df-slot 17367  df-ndx 17379  df-base 17395  df-hom 17459  df-cco 17460  df-homf 17851  df-comf 17852  df-oppc 17893

This theorem is used by:  yonpropd  18449  lmdpropd  50947