Theorem oppcinv 16825

Description: An inverse in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
oppcsect.b	⊢ 𝐵 = (Base‘𝐶)
oppcsect.o	⊢ 𝑂 = (oppCat‘𝐶)
oppcsect.c	⊢ (𝜑 → 𝐶 ∈ Cat)
oppcsect.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
oppcsect.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
oppcinv.s	⊢ 𝐼 = (Inv‘𝐶)
oppcinv.t	⊢ 𝐽 = (Inv‘𝑂)

Assertion

Ref	Expression
oppcinv	⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))

Proof of Theorem oppcinv

Step	Hyp	Ref	Expression
1		incom 4027	. . 3 ⊢ ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌))
2		oppcsect.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐶)
3		oppcsect.o	. . . . . . 7 ⊢ 𝑂 = (oppCat‘𝐶)
4		oppcsect.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		oppcsect.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		oppcsect.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		eqid 2777	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8		eqid 2777	. . . . . . 7 ⊢ (Sect‘𝑂) = (Sect‘𝑂)
9	2, 3, 4, 5, 6, 7, 8	oppcsect2 16824	. . . . . 6 ⊢ (𝜑 → (𝑌(Sect‘𝑂)𝑋) = ◡(𝑌(Sect‘𝐶)𝑋))
10	9	cnveqd 5543	. . . . 5 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = ◡◡(𝑌(Sect‘𝐶)𝑋))
11		eqid 2777	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2777	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
13		eqid 2777	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
14	2, 11, 12, 13, 7, 4, 5, 6	sectss 16797	. . . . . . 7 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
15		relxp 5373	. . . . . . 7 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
16		relss 5454	. . . . . . 7 ⊢ ((𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → (Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)) → Rel (𝑌(Sect‘𝐶)𝑋)))
17	14, 15, 16	mpisyl 21	. . . . . 6 ⊢ (𝜑 → Rel (𝑌(Sect‘𝐶)𝑋))
18		dfrel2 5837	. . . . . 6 ⊢ (Rel (𝑌(Sect‘𝐶)𝑋) ↔ ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
19	17, 18	sylib 210	. . . . 5 ⊢ (𝜑 → ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
20	10, 19	eqtrd 2813	. . . 4 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
21	2, 3, 4, 6, 5, 7, 8	oppcsect2 16824	. . . 4 ⊢ (𝜑 → (𝑋(Sect‘𝑂)𝑌) = ◡(𝑋(Sect‘𝐶)𝑌))
22	20, 21	ineq12d 4037	. . 3 ⊢ (𝜑 → (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
23	1, 22	syl5eq 2825	. 2 ⊢ (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
24	3, 2	oppcbas 16763	. . 3 ⊢ 𝐵 = (Base‘𝑂)
25		oppcinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝑂)
26	3	oppccat 16767	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
27	4, 26	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
28	24, 25, 27, 6, 5, 8	invfval 16804	. 2 ⊢ (𝜑 → (𝑋𝐽𝑌) = ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)))
29		oppcinv.s	. . 3 ⊢ 𝐼 = (Inv‘𝐶)
30	2, 29, 4, 5, 6, 7	invfval 16804	. 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
31	23, 28, 30	3eqtr4d 2823	1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))

Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2106   ∩ cin 3790   ⊆ wss 3791   × cxp 5353  ^◡ccnv 5354  Rel wrel 5360  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  Hom chom 16349  compcco 16350  Catccat 16710  Idccid 16711  oppCatcoppc 16756  Sectcsect 16789  Invcinv 16790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-tpos 7634  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-hom 16362  df-cco 16363  df-cat 16714  df-cid 16715  df-oppc 16757  df-sect 16792  df-inv 16793

This theorem is referenced by:  oppciso  16826  episect  16830