Theorem oppcinv 17916

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 4154	. . 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 2760	. . . . . . 7 ⊢ (Sect‘𝐶) = (Sect‘𝐶)
8		eqid 2760	. . . . . . 7 ⊢ (Sect‘𝑂) = (Sect‘𝑂)
9	2, 3, 4, 5, 6, 7, 8	oppcsect2 17915	. . . . . 6 ⊢ (𝜑 → (𝑌(Sect‘𝑂)𝑋) = ◡(𝑌(Sect‘𝐶)𝑋))
10	9	cnveqd 5849	. . . . 5 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = ◡◡(𝑌(Sect‘𝐶)𝑋))
11		eqid 2760	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqid 2760	. . . . . . . 8 ⊢ (comp‘𝐶) = (comp‘𝐶)
13		eqid 2760	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
14	2, 11, 12, 13, 7, 4, 5, 6	sectss 17888	. . . . . . 7 ⊢ (𝜑 → (𝑌(Sect‘𝐶)𝑋) ⊆ ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌)))
15		relxp 5665	. . . . . . 7 ⊢ Rel ((𝑌(Hom ‘𝐶)𝑋) × (𝑋(Hom ‘𝐶)𝑌))
16		relss 5754	. . . . . . 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 22	. . . . . 6 ⊢ (𝜑 → Rel (𝑌(Sect‘𝐶)𝑋))
18		dfrel2 6176	. . . . . 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 2795	. . . 4 ⊢ (𝜑 → ◡(𝑌(Sect‘𝑂)𝑋) = (𝑌(Sect‘𝐶)𝑋))
21	2, 3, 4, 6, 5, 7, 8	oppcsect2 17915	. . . 4 ⊢ (𝜑 → (𝑋(Sect‘𝑂)𝑌) = ◡(𝑋(Sect‘𝐶)𝑌))
22	20, 21	ineq12d 4166	. . 3 ⊢ (𝜑 → (◡(𝑌(Sect‘𝑂)𝑋) ∩ (𝑋(Sect‘𝑂)𝑌)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
23	1, 22	eqtrid 2807	. 2 ⊢ (𝜑 → ((𝑋(Sect‘𝑂)𝑌) ∩ ◡(𝑌(Sect‘𝑂)𝑋)) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
24	3, 2	oppcbas 17853	. . 3 ⊢ 𝐵 = (Base‘𝑂)
25		oppcinv.t	. . 3 ⊢ 𝐽 = (Inv‘𝑂)
26	3	oppccat 17857	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
27	4, 26	syl 18	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
28	24, 25, 27, 6, 5, 8	invfval 17895	. 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 17895	. 2 ⊢ (𝜑 → (𝑌𝐼𝑋) = ((𝑌(Sect‘𝐶)𝑋) ∩ ◡(𝑋(Sect‘𝐶)𝑌)))
31	23, 28, 30	3eqtr4d 2805	1 ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝑌𝐼𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∩ cin 3897   ⊆ wss 3898   × cxp 5645  ^◡ccnv 5646  Rel wrel 5652  ‘cfv 6527  (class class class)co 7408  Basecbs 17348  Hom chom 17400  compcco 17401  Catccat 17799  Idccid 17800  oppCatcoppc 17846  Sectcsect 17880  Invcinv 17881

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8221  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-nn 12305  df-2 12374  df-3 12375  df-4 12376  df-5 12377  df-6 12378  df-7 12379  df-8 12380  df-9 12381  df-n0 12576  df-z 12663  df-dec 12784  df-sets 17303  df-slot 17321  df-ndx 17333  df-base 17349  df-hom 17413  df-cco 17414  df-cat 17803  df-cid 17804  df-oppc 17847  df-sect 17883  df-inv 17884

This theorem is used by:  oppciso  17917  episect  17921