Theorem oppcinv 17722

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 4168	. . 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 2729	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8		eqid 2729	. . . . . . 7 ⊢ (Sect‘𝑂) = (Sect‘𝑂)
9	2, 3, 4, 5, 6, 7, 8	oppcsect2 17721	. . . . . 6 ⊢ (𝜑 → (𝑌(Sect‘𝑂)𝑋) = ◡(𝑌(Sect‘𝐶)𝑋))
10	9	cnveqd 5829	. . . . 5 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = ◡◡(𝑌(Sect‘𝐶)𝑋))
11		eqid 2729	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2729	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
13		eqid 2729	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
14	2, 11, 12, 13, 7, 4, 5, 6	sectss 17694	. . . . . . 7 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
15		relxp 5649	. . . . . . 7 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
16		relss 5736	. . . . . . 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 6150	. . . . . 6 ⊢ (Rel (𝑌(Sect‘𝐶)𝑋) ↔ ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
19	17, 18	sylib 218	. . . . 5 ⊢ (𝜑 → ◡◡(𝑌(Sect‘𝐶)𝑋) = (𝑌(Sect‘𝐶)𝑋))
20	10, 19	eqtrd 2764	. . . 4 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
21	2, 3, 4, 6, 5, 7, 8	oppcsect2 17721	. . . 4 ⊢ (𝜑 → (𝑋(Sect‘𝑂)𝑌) = ◡(𝑋(Sect‘𝐶)𝑌))
22	20, 21	ineq12d 4180	. . 3 ⊢ (𝜑 → (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
23	1, 22	eqtrid 2776	. 2 ⊢ (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
24	3, 2	oppcbas 17659	. . 3 ⊢ 𝐵 = (Base‘𝑂)
25		oppcinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝑂)
26	3	oppccat 17663	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
27	4, 26	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
28	24, 25, 27, 6, 5, 8	invfval 17701	. 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 17701	. 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
31	23, 28, 30	3eqtr4d 2774	1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ∩ cin 3910   ⊆ wss 3911   × cxp 5629  ^◡ccnv 5630  Rel wrel 5636  ‘cfv 6499  (class class class)co 7369  Basecbs 17155  Hom chom 17207  compcco 17208  Catccat 17605  Idccid 17606  oppCatcoppc 17652  Sectcsect 17686  Invcinv 17687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-tpos 8182  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-hom 17220  df-cco 17221  df-cat 17609  df-cid 17610  df-oppc 17653  df-sect 17689  df-inv 17690

This theorem is referenced by:  oppciso  17723  episect  17727