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Mirrors > Home > MPE Home > Th. List > sectss | Structured version Visualization version GIF version |
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 16882 | . 2 ⊢ (𝜑 → (𝑋𝑆𝑌) = {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))}) |
10 | opabssxp 5494 | . 2 ⊢ {〈𝑓, 𝑔〉 ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(〈𝑋, 𝑌〉 · 𝑋)𝑓) = ( 1 ‘𝑋))} ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) | |
11 | 9, 10 | syl6eqss 3913 | 1 ⊢ (𝜑 → (𝑋𝑆𝑌) ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3831 〈cop 4448 {copab 4992 × cxp 5406 ‘cfv 6190 (class class class)co 6978 Basecbs 16342 Hom chom 16435 compcco 16436 Catccat 16796 Idccid 16797 Sectcsect 16875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-ov 6981 df-oprab 6982 df-mpo 6983 df-1st 7503 df-2nd 7504 df-sect 16878 |
This theorem is referenced by: isinv 16891 invss 16892 oppcsect2 16910 oppcinv 16911 |
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