Theorem oppcsect 17796

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 2736	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
3		oppcsect.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
4		oppcsect.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝑋 ∈ 𝐵)
6		oppcsect.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	6	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝑌 ∈ 𝐵)
8	1, 2, 3, 5, 7, 5	oppcco 17734	. . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
11		eqid 2736	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3, 11	oppcid 17738	. . . . . . 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 6883	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
15	8, 14	eqeq12d 2752	. . . 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 2736	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19	18, 3	oppchom 17732	. . . . . . 7 ⊢ (𝑋(Hom ‘𝑂)𝑌) = (𝑌(Hom ‘𝐶)𝑋)
20	19	eleq2i 2827	. . . . . 6 ⊢ (𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ↔ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
21	18, 3	oppchom 17732	. . . . . . 7 ⊢ (𝑌(Hom ‘𝑂)𝑋) = (𝑋(Hom ‘𝐶)𝑌)
22	21	eleq2i 2827	. . . . . 6 ⊢ (𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋) ↔ 𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌))
23	20, 22	anbi12ci 629	. . . . 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 275	. . 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 17735	. . 3 ⊢ 𝐵 = (Base‘𝑂)
29		eqid 2736	. . 3 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
30		eqid 2736	. . 3 ⊢ (comp‘𝑂) = (comp‘𝑂)
31		eqid 2736	. . 3 ⊢ (Id‘𝑂) = (Id‘𝑂)
32		oppcsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝑂)
33	3	oppccat 17739	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
34	9, 33	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
35	28, 29, 30, 31, 32, 34, 4, 6	issect 17771	. 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 17771	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4612   class class class wbr 5124  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  Hom chom 17287  compcco 17288  Catccat 17681  Idccid 17682  oppCatcoppc 17728  Sectcsect 17762

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 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-sets 17188  df-slot 17206  df-ndx 17218  df-base 17234  df-hom 17300  df-cco 17301  df-cat 17685  df-cid 17686  df-oppc 17729  df-sect 17765

This theorem is referenced by:  oppcsect2  17797  sectepi  17802  episect  17803