Theorem 2oppccomf 16843

Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 16855. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypothesis

Ref	Expression
oppcbas.1	⊢ 𝑂 = (oppCat‘𝐶)

Assertion

Ref	Expression
2oppccomf	⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))

Proof of Theorem 2oppccomf

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

Step	Hyp	Ref	Expression
1		oppcbas.1	. . . . . . . . 9 ⊢ 𝑂 = (oppCat‘𝐶)
2		eqid 2772	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 16836	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝑂)
4		eqid 2772	. . . . . . . 8 ⊢ (comp‘𝑂) = (comp‘𝑂)
5		eqid 2772	. . . . . . . 8 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
6		simpr1 1174	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
7		simpr2 1175	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
8		simpr3 1176	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
9	3, 4, 5, 6, 7, 8	oppcco 16835	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓) = (𝑓(⟨𝑧, 𝑦⟩(comp‘𝑂)𝑥)𝑔))
10		eqid 2772	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
11	2, 10, 1, 8, 7, 6	oppcco 16835	. . . . . . 7 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑓(⟨𝑧, 𝑦⟩(comp‘𝑂)𝑥)𝑔) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓))
12	9, 11	eqtr2d 2809	. . . . . 6 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓))
13	12	ralrimivw 3127	. . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓))
14	13	ralrimivw 3127	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓))
15	14	ralrimivvva 3136	. . 3 ⊢ (⊤ → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓))
16		eqid 2772	. . . 4 ⊢ (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂))
17		eqid 2772	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
18		eqidd 2773	. . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘𝐶))
19	1, 2	2oppcbas 16841	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝑂))
20	19	a1i 11	. . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂)))
21	1	2oppchomf 16842	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
22	21	a1i 11	. . . 4 ⊢ (⊤ → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)))
23	10, 16, 17, 18, 20, 22	comfeq 16824	. . 3 ⊢ (⊤ → ((compf‘𝐶) = (compf‘(oppCat‘𝑂)) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(oppCat‘𝑂))𝑧)𝑓)))
24	15, 23	mpbird 249	. 2 ⊢ (⊤ → (compf‘𝐶) = (compf‘(oppCat‘𝑂)))
25	24	mptru 1514	1 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))

Syntax hints:   ∧ wa 387   ∧ w3a 1068   = wceq 1507  ⊤wtru 1508   ∈ wcel 2048  ∀wral 3082  ⟨cop 4441  ‘cfv 6182  (class class class)co 6970  Basecbs 16329  Hom chom 16422  compcco 16423  Hom_f chomf 16785  comp_fccomf 16786  oppCatcoppc 16829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-tpos 7688  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-er 8081  df-en 8299  df-dom 8300  df-sdom 8301  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-nn 11432  df-2 11496  df-3 11497  df-4 11498  df-5 11499  df-6 11500  df-7 11501  df-8 11502  df-9 11503  df-n0 11701  df-z 11787  df-dec 11905  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-hom 16435  df-cco 16436  df-homf 16789  df-comf 16790  df-oppc 16830

This theorem is referenced by:  oppcepi  16857  oppchofcl  17358  oppcyon  17367  oyoncl  17368