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| Description: The section relation is a relation between morphisms from 𝑋 to 𝑌 and morphisms from 𝑌 to 𝑋. (Contributed by Mario Carneiro, 2-Jan-2017.) | 
| Ref | Expression | 
|---|---|
| issect.b | ⊢ 𝐵 = (Base‘𝐶) | 
| issect.h | ⊢ 𝐻 = (Hom ‘𝐶) | 
| issect.o | ⊢ · = (comp‘𝐶) | 
| issect.i | ⊢ 1 = (Id‘𝐶) | 
| issect.s | ⊢ 𝑆 = (Sect‘𝐶) | 
| issect.c | ⊢ (𝜑 → 𝐶 ∈ Cat) | 
| issect.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| issect.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| sectss | ⊢ (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | issect.b | . . 3 ⊢ 𝐵 = (Base‘𝐶) | |
| 2 | issect.h | . . 3 ⊢ 𝐻 = (Hom ‘𝐶) | |
| 3 | issect.o | . . 3 ⊢ · = (comp‘𝐶) | |
| 4 | issect.i | . . 3 ⊢ 1 = (Id‘𝐶) | |
| 5 | issect.s | . . 3 ⊢ 𝑆 = (Sect‘𝐶) | |
| 6 | issect.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
| 7 | issect.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 8 | issect.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 9 | 1, 2, 3, 4, 5, 6, 7, 8 | sectfval 17796 | . 2 ⊢ (𝜑 → (𝑋𝑆𝑌) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}) | 
| 10 | opabssxp 5777 | . 2 ⊢ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))} ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) | |
| 11 | 9, 10 | eqsstrdi 4027 | 1 ⊢ (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 〈cop 4631 {copab 5204 × cxp 5682 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Hom chom 17309 compcco 17310 Catccat 17708 Idccid 17709 Sectcsect 17789 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-sect 17792 | 
| This theorem is referenced by: isinv 17805 invss 17806 oppcsect2 17824 oppcinv 17825 | 
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