Theorem oppcsect 17488

Description: A section 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)
oppcsect.s	⊢ 𝑆 = (Sect‘𝐶)
oppcsect.t	⊢ 𝑇 = (Sect‘𝑂)

Assertion

Ref	Expression
oppcsect	⊢ (𝜑 → (𝐹(𝑋𝑇𝑌)𝐺 ↔ 𝐺(𝑋𝑆𝑌)𝐹))

Proof of Theorem oppcsect

Step	Hyp	Ref	Expression
1		oppcsect.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
2		eqid 2738	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
3		oppcsect.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
4		oppcsect.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝑋 ∈ 𝐵)
6		oppcsect.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	6	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝑌 ∈ 𝐵)
8	1, 2, 3, 5, 7, 5	oppcco 17425	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺))
9		oppcsect.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
10	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
11		eqid 2738	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3, 11	oppcid 17430	. . . . . . 7 ⊢ (𝐶 ∈ Cat → (Id‘𝑂) = (Id‘𝐶))
13	10, 12	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → (Id‘𝑂) = (Id‘𝐶))
14	13	fveq1d 6778	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
15	8, 14	eqeq12d 2754	. . . 4 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → ((𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋) ↔ (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)))
16	15	pm5.32da 579	. . 3 ⊢ (𝜑 → (((𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋)) ↔ ((𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))))
17		df-3an 1088	. . . 4 ⊢ ((𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋)) ↔ ((𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋)))
18		eqid 2738	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19	18, 3	oppchom 17423	. . . . . . 7 ⊢ (𝑋(Hom ‘𝑂)𝑌) = (𝑌(Hom ‘𝐶)𝑋)
20	19	eleq2i 2830	. . . . . 6 ⊢ (𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ↔ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
21	18, 3	oppchom 17423	. . . . . . 7 ⊢ (𝑌(Hom ‘𝑂)𝑋) = (𝑋(Hom ‘𝐶)𝑌)
22	21	eleq2i 2830	. . . . . 6 ⊢ (𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋) ↔ 𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌))
23	20, 22	anbi12ci 628	. . . . 5 ⊢ ((𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋)) ↔ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)))
24	23	anbi1i 624	. . . 4 ⊢ (((𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋)) ↔ ((𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋)))
25	17, 24	bitri 274	. . 3 ⊢ ((𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋)) ↔ ((𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋)))
26		df-3an 1088	. . 3 ⊢ ((𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)) ↔ ((𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)) ∧ (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋)))
27	16, 25, 26	3bitr4g 314	. 2 ⊢ (𝜑 → ((𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋)) ↔ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))))
28	3, 1	oppcbas 17426	. . 3 ⊢ 𝐵 = (Base‘𝑂)
29		eqid 2738	. . 3 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
30		eqid 2738	. . 3 ⊢ (comp‘𝑂) = (comp‘𝑂)
31		eqid 2738	. . 3 ⊢ (Id‘𝑂) = (Id‘𝑂)
32		oppcsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝑂)
33	3	oppccat 17431	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
34	9, 33	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
35	28, 29, 30, 31, 32, 34, 4, 6	issect 17463	. 2 ⊢ (𝜑 → (𝐹(𝑋𝑇𝑌)𝐺 ↔ (𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋) ∧ (𝐺(⟨𝑋, 𝑌⟩(comp‘𝑂)𝑋)𝐹) = ((Id‘𝑂)‘𝑋))))
36		oppcsect.s	. . 3 ⊢ 𝑆 = (Sect‘𝐶)
37	1, 18, 2, 11, 36, 9, 4, 6	issect 17463	. 2 ⊢ (𝜑 → (𝐺(𝑋𝑆𝑌)𝐹 ↔ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋) ∧ (𝐹(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑋)𝐺) = ((Id‘𝐶)‘𝑋))))
38	27, 35, 37	3bitr4d 311	1 ⊢ (𝜑 → (𝐹(𝑋𝑇𝑌)𝐺 ↔ 𝐺(𝑋𝑆𝑌)𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ⟨cop 4569   class class class wbr 5076  ‘cfv 6435  (class class class)co 7277  Basecbs 16910  Hom chom 16971  compcco 16972  Catccat 17371  Idccid 17372  oppCatcoppc 17418  Sectcsect 17454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5211  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588  ax-cnex 10925  ax-resscn 10926  ax-1cn 10927  ax-icn 10928  ax-addcl 10929  ax-addrcl 10930  ax-mulcl 10931  ax-mulrcl 10932  ax-mulcom 10933  ax-addass 10934  ax-mulass 10935  ax-distr 10936  ax-i2m1 10937  ax-1ne0 10938  ax-1rid 10939  ax-rnegex 10940  ax-rrecex 10941  ax-cnre 10942  ax-pre-lttri 10943  ax-pre-lttrn 10944  ax-pre-ltadd 10945  ax-pre-mulgt0 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-tr 5194  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-pred 6204  df-ord 6271  df-on 6272  df-lim 6273  df-suc 6274  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-riota 7234  df-ov 7280  df-oprab 7281  df-mpo 7282  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8040  df-frecs 8095  df-wrecs 8126  df-recs 8200  df-rdg 8239  df-er 8496  df-en 8732  df-dom 8733  df-sdom 8734  df-pnf 11009  df-mnf 11010  df-xr 11011  df-ltxr 11012  df-le 11013  df-sub 11205  df-neg 11206  df-nn 11972  df-2 12034  df-3 12035  df-4 12036  df-5 12037  df-6 12038  df-7 12039  df-8 12040  df-9 12041  df-n0 12232  df-z 12318  df-dec 12436  df-sets 16863  df-slot 16881  df-ndx 16893  df-base 16911  df-hom 16984  df-cco 16985  df-cat 17375  df-cid 17376  df-oppc 17419  df-sect 17457

This theorem is referenced by:  oppcsect2  17489  sectepi  17494  episect  17495