Theorem 2oppccomf 17806

Description: The double opposite category has the same composition as the original category. Intended for use with property lemmas such as monpropd 17819. (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 2766	. . . . . . . . 9 ⊢ (Base‘𝐶) = (Base‘𝐶)
3	1, 2	oppcbas 17799	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝑂)
4		eqid 2766	. . . . . . . 8 ⊢ (comp‘𝑂) = (comp‘𝑂)
5		eqid 2766	. . . . . . . 8 ⊢ (oppCat‘𝑂) = (oppCat‘𝑂)
6		simpr1 1213	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑥 ∈ (Base‘𝐶))
7		simpr2 1214	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑦 ∈ (Base‘𝐶))
8		simpr3 1215	. . . . . . . 8 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶) ∧ 𝑧 ∈ (Base‘𝐶))) → 𝑧 ∈ (Base‘𝐶))
9	3, 4, 5, 6, 7, 8	oppcco 17798	. . . . . . 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 2766	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
11	2, 10, 1, 8, 7, 6	oppcco 17798	. . . . . . 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 2802	. . . . . 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 3164	. . . . 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 3164	. . . 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 3214	. . 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 2766	. . . 4 ⊢ (comp‘(oppCat‘𝑂)) = (comp‘(oppCat‘𝑂))
17		eqid 2766	. . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
18		eqidd 2767	. . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘𝐶))
19	1, 2	2oppcbas 17804	. . . . 5 ⊢ (Base‘𝐶) = (Base‘(oppCat‘𝑂))
20	19	a1i 11	. . . 4 ⊢ (⊤ → (Base‘𝐶) = (Base‘(oppCat‘𝑂)))
21	1	2oppchomf 17805	. . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂))
22	21	a1i 11	. . . 4 ⊢ (⊤ → (Homf ‘𝐶) = (Homf ‘(oppCat‘𝑂)))
23	10, 16, 17, 18, 20, 22	comfeq 17787	. . 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 260	. 2 ⊢ (⊤ → (compf‘𝐶) = (compf‘(oppCat‘𝑂)))
25	24	mptru 1577	1 ⊢ (compf‘𝐶) = (compf‘(oppCat‘𝑂))

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ⊤wtru 1571   ∈ wcel 2146  ∀wral 3082  ⟨cop 4600  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  Hom chom 17346  compcco 17347  Hom_f chomf 17747  comp_fccomf 17748  oppCatcoppc 17792

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745  ax-cnex 11174  ax-resscn 11175  ax-1cn 11176  ax-icn 11177  ax-addcl 11178  ax-addrcl 11179  ax-mulcl 11180  ax-mulrcl 11181  ax-mulcom 11182  ax-addass 11183  ax-mulass 11184  ax-distr 11185  ax-i2m1 11186  ax-1ne0 11187  ax-1rid 11188  ax-rnegex 11189  ax-rrecex 11190  ax-cnre 11191  ax-pre-lttri 11192  ax-pre-lttrn 11193  ax-pre-ltadd 11194  ax-pre-mulgt0 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-nel 3068  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-om 7872  df-1st 7995  df-2nd 7996  df-tpos 8231  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-rdg 8406  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11461  df-neg 11462  df-nn 12252  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-7 12326  df-8 12327  df-9 12328  df-n0 12523  df-z 12610  df-dec 12730  df-sets 17249  df-slot 17267  df-ndx 17279  df-base 17295  df-hom 17359  df-cco 17360  df-homf 17751  df-comf 17752  df-oppc 17793

This theorem is used by:  oppcepi  17821  oppchofcl  18341  oppcyon  18350  oyoncl  18351  oppccatb  49835  oppccicb  49870  funcoppc2  49962  natoppfb  50050  cmddu  50487  termolmd  50489