Theorem oppcinv 17706

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 4161	. . 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 2736	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8		eqid 2736	. . . . . . 7 ⊢ (Sect‘𝑂) = (Sect‘𝑂)
9	2, 3, 4, 5, 6, 7, 8	oppcsect2 17705	. . . . . 6 ⊢ (𝜑 → (𝑌(Sect‘𝑂)𝑋) = ◡(𝑌(Sect‘𝐶)𝑋))
10	9	cnveqd 5824	. . . . 5 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = ◡◡(𝑌(Sect‘𝐶)𝑋))
11		eqid 2736	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2736	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
13		eqid 2736	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
14	2, 11, 12, 13, 7, 4, 5, 6	sectss 17678	. . . . . . 7 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
15		relxp 5642	. . . . . . 7 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
16		relss 5731	. . . . . . 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 6147	. . . . . 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 2771	. . . 4 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
21	2, 3, 4, 6, 5, 7, 8	oppcsect2 17705	. . . 4 ⊢ (𝜑 → (𝑋(Sect‘𝑂)𝑌) = ◡(𝑋(Sect‘𝐶)𝑌))
22	20, 21	ineq12d 4173	. . 3 ⊢ (𝜑 → (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
23	1, 22	eqtrid 2783	. 2 ⊢ (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
24	3, 2	oppcbas 17643	. . 3 ⊢ 𝐵 = (Base‘𝑂)
25		oppcinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝑂)
26	3	oppccat 17647	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
27	4, 26	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
28	24, 25, 27, 6, 5, 8	invfval 17685	. 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 17685	. 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
31	23, 28, 30	3eqtr4d 2781	1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ∩ cin 3900   ⊆ wss 3901   × cxp 5622  ^◡ccnv 5623  Rel wrel 5629  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  Hom chom 17190  compcco 17191  Catccat 17589  Idccid 17590  oppCatcoppc 17636  Sectcsect 17670  Invcinv 17671

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-1st 7933  df-2nd 7934  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-3 12211  df-4 12212  df-5 12213  df-6 12214  df-7 12215  df-8 12216  df-9 12217  df-n0 12404  df-z 12491  df-dec 12610  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-hom 17203  df-cco 17204  df-cat 17593  df-cid 17594  df-oppc 17637  df-sect 17673  df-inv 17674

This theorem is referenced by:  oppciso  17707  episect  17711