Theorem oppccomfpropd 17688

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 2729	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2729	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2729	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2729	. . . . . 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 2729	. . . . . . . 8 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
14	2, 13	oppchom 17676	. . . . . . 7 ⊢ (𝑦(Hom ‘(oppCat‘𝐶))𝑧) = (𝑧(Hom ‘𝐶)𝑦)
15	12, 14	eleqtrdi 2838	. . . . . 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 17676	. . . . . . 7 ⊢ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) = (𝑦(Hom ‘𝐶)𝑥)
18	16, 17	eleqtrdi 2838	. . . . . 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 17669	. . . . 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 17678	. . . . 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 2729	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
22		eqid 2729	. . . . . 6 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
23	5	homfeqbas 17657	. . . . . . . 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 2830	. . . . . 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 2830	. . . . . 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 2830	. . . . . 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 17678	. . . . 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 2774	. . . 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 3180	. . 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 3183	. 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 2729	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33		eqid 2729	. . 3 ⊢ (comp‘(oppCat‘𝐷)) = (comp‘(oppCat‘𝐷))
34		eqid 2729	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
35	13, 1	oppcbas 17679	. . . 4 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
36	35	a1i 11	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐶)))
37	22, 21	oppcbas 17679	. . . 4 ⊢ (Base‘𝐷) = (Base‘(oppCat‘𝐷))
38	23, 37	eqtrdi 2780	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐷)))
39	5	oppchomfpropd 17687	. . 3 ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷)))
40	32, 33, 34, 36, 38, 39	comfeq 17667	. 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 3044  ⟨cop 4595  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  Hom chom 17231  compcco 17232  Hom_f chomf 17627  comp_fccomf 17628  oppCatcoppc 17672

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 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-hom 17244  df-cco 17245  df-homf 17631  df-comf 17632  df-oppc 17673

This theorem is referenced by:  yonpropd  18229  lmdpropd  49643