Theorem oppcsect 16645

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 2771	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
3		oppcsect.o	. . . . . 6 ⊢ 𝑂 = (oppCat‘𝐶)
4		oppcsect.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5	4	adantr 466	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝑋 ∈ 𝐵)
6		oppcsect.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	6	adantr 466	. . . . . 6 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝑌 ∈ 𝐵)
8	1, 2, 3, 5, 7, 5	oppcco 16584	. . . . 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 466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → 𝐶 ∈ Cat)
11		eqid 2771	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3, 11	oppcid 16588	. . . . . . 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 6334	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
15	8, 14	eqeq12d 2786	. . . 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 568	. . 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 1073	. . . 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 2771	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19	18, 3	oppchom 16582	. . . . . . 7 ⊢ (𝑋(Hom ‘𝑂)𝑌) = (𝑌(Hom ‘𝐶)𝑋)
20	19	eleq2i 2842	. . . . . 6 ⊢ (𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ↔ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
21	18, 3	oppchom 16582	. . . . . . 7 ⊢ (𝑌(Hom ‘𝑂)𝑋) = (𝑋(Hom ‘𝐶)𝑌)
22	21	eleq2i 2842	. . . . . 6 ⊢ (𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋) ↔ 𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌))
23	20, 22	anbi12ci 613	. . . . 5 ⊢ ((𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ∧ 𝐺 ∈ (𝑌(Hom ‘𝑂)𝑋)) ↔ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋)))
24	23	anbi1i 610	. . . 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 264	. . 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 1073	. . 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 303	. 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 16585	. . 3 ⊢ 𝐵 = (Base‘𝑂)
29		eqid 2771	. . 3 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
30		eqid 2771	. . 3 ⊢ (comp‘𝑂) = (comp‘𝑂)
31		eqid 2771	. . 3 ⊢ (Id‘𝑂) = (Id‘𝑂)
32		oppcsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝑂)
33	3	oppccat 16589	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
34	9, 33	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
35	28, 29, 30, 31, 32, 34, 4, 6	issect 16620	. 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 16620	. 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 300	1 ⊢ (𝜑 → (𝐹(𝑋𝑇𝑌)𝐺 ↔ 𝐺(𝑋𝑆𝑌)𝐹))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ⟨cop 4322   class class class wbr 4786  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  Idccid 16533  oppCatcoppc 16578  Sectcsect 16611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-tpos 7504  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-9 11288  df-n0 11495  df-z 11580  df-dec 11696  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-hom 16174  df-cco 16175  df-cat 16536  df-cid 16537  df-oppc 16579  df-sect 16614

This theorem is referenced by:  oppcsect2  16646  sectepi  16651  episect  16652