Theorem oppcinv 17155

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 4091	. . 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 2738	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8		eqid 2738	. . . . . . 7 ⊢ (Sect‘𝑂) = (Sect‘𝑂)
9	2, 3, 4, 5, 6, 7, 8	oppcsect2 17154	. . . . . 6 ⊢ (𝜑 → (𝑌(Sect‘𝑂)𝑋) = ◡(𝑌(Sect‘𝐶)𝑋))
10	9	cnveqd 5718	. . . . 5 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = ◡◡(𝑌(Sect‘𝐶)𝑋))
11		eqid 2738	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2738	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
13		eqid 2738	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
14	2, 11, 12, 13, 7, 4, 5, 6	sectss 17127	. . . . . . 7 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
15		relxp 5543	. . . . . . 7 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
16		relss 5627	. . . . . . 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 6021	. . . . . 6 ⊢ (Rel (𝑌(Sect‘𝐶)𝑋) ↔ ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
19	17, 18	sylib 221	. . . . 5 ⊢ (𝜑 → ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
20	10, 19	eqtrd 2773	. . . 4 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
21	2, 3, 4, 6, 5, 7, 8	oppcsect2 17154	. . . 4 ⊢ (𝜑 → (𝑋(Sect‘𝑂)𝑌) = ◡(𝑋(Sect‘𝐶)𝑌))
22	20, 21	ineq12d 4104	. . 3 ⊢ (𝜑 → (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
23	1, 22	syl5eq 2785	. 2 ⊢ (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
24	3, 2	oppcbas 17092	. . 3 ⊢ 𝐵 = (Base‘𝑂)
25		oppcinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝑂)
26	3	oppccat 17096	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
27	4, 26	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
28	24, 25, 27, 6, 5, 8	invfval 17134	. 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 17134	. 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
31	23, 28, 30	3eqtr4d 2783	1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∩ cin 3842   ⊆ wss 3843   × cxp 5523  ^◡ccnv 5524  Rel wrel 5530  ‘cfv 6339  (class class class)co 7170  Basecbs 16586  Hom chom 16679  compcco 16680  Catccat 17038  Idccid 17039  oppCatcoppc 17085  Sectcsect 17119  Invcinv 17120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-tpos 7921  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-7 11784  df-8 11785  df-9 11786  df-n0 11977  df-z 12063  df-dec 12180  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-hom 16692  df-cco 16693  df-cat 17042  df-cid 17043  df-oppc 17086  df-sect 17122  df-inv 17123

This theorem is referenced by:  oppciso  17156  episect  17160