Theorem oppccomfpropd 17684

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 2737	. . . . . 6 ⊢ (Base‘𝐶) = (Base‘𝐶)
2		eqid 2737	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
3		eqid 2737	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
4		eqid 2737	. . . . . 6 ⊢ (comp‘𝐷) = (comp‘𝐷)
5		oppchomfpropd.1	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
6	5	ad2antrr 727	. . . . . 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 727	. . . . . 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 1219	. . . . . 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 1218	. . . . . 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 1217	. . . . . 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 773	. . . . . . 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 2737	. . . . . . . 8 ⊢ (oppCat‘𝐶) = (oppCat‘𝐶)
14	2, 13	oppchom 17672	. . . . . . 7 ⊢ (𝑦(Hom ‘(oppCat‘𝐶))𝑧) = (𝑧(Hom ‘𝐶)𝑦)
15	12, 14	eleqtrdi 2847	. . . . . 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 771	. . . . . . 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 17672	. . . . . . 7 ⊢ (𝑥(Hom ‘(oppCat‘𝐶))𝑦) = (𝑦(Hom ‘𝐶)𝑥)
18	16, 17	eleqtrdi 2847	. . . . . 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 17665	. . . . 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 17674	. . . . 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 2737	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
22		eqid 2737	. . . . . 6 ⊢ (oppCat‘𝐷) = (oppCat‘𝐷)
23	5	homfeqbas 17653	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
24	23	ad2antrr 727	. . . . . . 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 2839	. . . . . 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 2839	. . . . . 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 2839	. . . . . 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 17674	. . . . 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 2782	. . . 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 3181	. . 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 3184	. 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 2737	. . 3 ⊢ (comp‘(oppCat‘𝐶)) = (comp‘(oppCat‘𝐶))
33		eqid 2737	. . 3 ⊢ (comp‘(oppCat‘𝐷)) = (comp‘(oppCat‘𝐷))
34		eqid 2737	. . 3 ⊢ (Hom ‘(oppCat‘𝐶)) = (Hom ‘(oppCat‘𝐶))
35	13, 1	oppcbas 17675	. . . 4 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝐶))
36	35	a1i 11	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐶)))
37	22, 21	oppcbas 17675	. . . 4 ⊢ (Base‘𝐷) = (Base‘(oppCat‘𝐷))
38	23, 37	eqtrdi 2788	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘(oppCat‘𝐷)))
39	5	oppchomfpropd 17683	. . 3 ⊢ (𝜑 → (Homf ‘(oppCat‘𝐶)) = (Homf ‘(oppCat‘𝐷)))
40	32, 33, 34, 36, 38, 39	comfeq 17663	. 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 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ⟨cop 4574  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  Hom chom 17222  compcco 17223  Hom_f chomf 17623  comp_fccomf 17624  oppCatcoppc 17668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-hom 17235  df-cco 17236  df-homf 17627  df-comf 17628  df-oppc 17669

This theorem is referenced by:  yonpropd  18225  lmdpropd  50144