Theorem 2oppccomf 17676

Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 17689. (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 2731	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17668	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝑂)
4		eqid 2731	. . . . . . . 8 ⊢ (comp‘𝑂) = (comp‘𝑂)
5		eqid 2731	. . . . . . . 8 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
6		simpr1 1193	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
7		simpr2 1194	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
8		simpr3 1195	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
9	3, 4, 5, 6, 7, 8	oppcco 17667	. . . . . . 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 2731	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
11	2, 10, 1, 8, 7, 6	oppcco 17667	. . . . . . 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 2772	. . . . . 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 3149	. . . . 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 3149	. . . 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 3202	. . 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 2731	. . . 4 ⊢ (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂))
17		eqid 2731	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
18		eqidd 2732	. . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘𝐶))
19	1, 2	2oppcbas 17674	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝑂))
20	19	a1i 11	. . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂)))
21	1	2oppchomf 17675	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
22	21	a1i 11	. . . 4 ⊢ (⊤ → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)))
23	10, 16, 17, 18, 20, 22	comfeq 17655	. . 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 256	. 2 ⊢ (⊤ → (compf‘𝐶) = (compf‘(oppCat‘𝑂)))
25	24	mptru 1547	1 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ⊤wtru 1541   ∈ wcel 2105  ∀wral 3060  ⟨cop 4635  ‘cfv 6544  (class class class)co 7412  Basecbs 17149  Hom chom 17213  compcco 17214  Hom_f chomf 17615  comp_fccomf 17616  oppCatcoppc 17660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-tpos 8214  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-hom 17226  df-cco 17227  df-homf 17619  df-comf 17620  df-oppc 17661

This theorem is referenced by:  oppcepi  17691  oppchofcl  18218  oppcyon  18227  oyoncl  18228