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Theorem oppcsect 17729

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 2732	. . . . . 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 17666	. . . . 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 2732	. . . . . . . 8 ⊢ (Id‘𝐶) = (Id‘𝐶)
12	3, 11	oppcid 17671	. . . . . . 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 6893	. . . . 5 ⊢ ((𝜑 ∧ (𝐺 ∈ (𝑋(Hom ‘𝐶)𝑌) ∧ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))) → ((Id‘𝑂)‘𝑋) = ((Id‘𝐶)‘𝑋))
15	8, 14	eqeq12d 2748	. . . 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 1089	. . . 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 2732	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
19	18, 3	oppchom 17664	. . . . . . 7 ⊢ (𝑋(Hom ‘𝑂)𝑌) = (𝑌(Hom ‘𝐶)𝑋)
20	19	eleq2i 2825	. . . . . 6 ⊢ (𝐹 ∈ (𝑋(Hom ‘𝑂)𝑌) ↔ 𝐹 ∈ (𝑌(Hom ‘𝐶)𝑋))
21	18, 3	oppchom 17664	. . . . . . 7 ⊢ (𝑌(Hom ‘𝑂)𝑋) = (𝑋(Hom ‘𝐶)𝑌)
22	21	eleq2i 2825	. . . . . 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 1089	. . 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 313	. 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 17667	. . 3 ⊢ 𝐵 = (Base‘𝑂)
29		eqid 2732	. . 3 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂)
30		eqid 2732	. . 3 ⊢ (comp‘𝑂) = (comp‘𝑂)
31		eqid 2732	. . 3 ⊢ (Id‘𝑂) = (Id‘𝑂)
32		oppcsect.t	. . 3 ⊢ 𝑇 = (Sect‘𝑂)
33	3	oppccat 17672	. . . 4 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat)
34	9, 33	syl 17	. . 3 ⊢ (𝜑 → 𝑂 ∈ Cat)
35	28, 29, 30, 31, 32, 34, 4, 6	issect 17704	. 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 17704	. 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 310	1 ⊢ (𝜑 → (𝐹(𝑋𝑇𝑌)𝐺 ↔ 𝐺(𝑋𝑆𝑌)𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⟨cop 4634 class class class wbr 5148 ‘cfv 6543 (class class class)co 7411 Basecbs 17148 Hom chom 17212 compcco 17213 Catccat 17612 Idccid 17613 oppCatcoppc 17659 Sectcsect 17695

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-hom 17225 df-cco 17226 df-cat 17616 df-cid 17617 df-oppc 17660 df-sect 17698

This theorem is referenced by: oppcsect2 17730 sectepi 17735 episect 17736

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