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Mirrors > Home > MPE Home > Th. List > oppcsect2 | Structured version Visualization version GIF version |
Description: A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.) |
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) |
Ref | Expression |
---|---|
oppcsect2 | ⊢ (𝜑 → (𝑋𝑇𝑌) = ◡(𝑋𝑆𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oppcsect.o | . . . . 5 ⊢ 𝑂 = (oppCat‘𝐶) | |
2 | oppcsect.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
3 | 1, 2 | oppcbas 17194 | . . . 4 ⊢ 𝐵 = (Base‘𝑂) |
4 | eqid 2734 | . . . 4 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂) | |
5 | eqid 2734 | . . . 4 ⊢ (comp‘𝑂) = (comp‘𝑂) | |
6 | eqid 2734 | . . . 4 ⊢ (Id‘𝑂) = (Id‘𝑂) | |
7 | oppcsect.t | . . . 4 ⊢ 𝑇 = (Sect‘𝑂) | |
8 | oppcsect.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
9 | 1 | oppccat 17198 | . . . . 5 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑂 ∈ Cat) |
11 | oppcsect.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | oppcsect.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
13 | 3, 4, 5, 6, 7, 10, 11, 12 | sectss 17229 | . . 3 ⊢ (𝜑 → (𝑋𝑇𝑌) ⊆ ((𝑋(Hom ‘𝑂)𝑌) × (𝑌(Hom ‘𝑂)𝑋))) |
14 | relxp 5558 | . . 3 ⊢ Rel ((𝑋(Hom ‘𝑂)𝑌) × (𝑌(Hom ‘𝑂)𝑋)) | |
15 | relss 5642 | . . 3 ⊢ ((𝑋𝑇𝑌) ⊆ ((𝑋(Hom ‘𝑂)𝑌) × (𝑌(Hom ‘𝑂)𝑋)) → (Rel ((𝑋(Hom ‘𝑂)𝑌) × (𝑌(Hom ‘𝑂)𝑋)) → Rel (𝑋𝑇𝑌))) | |
16 | 13, 14, 15 | mpisyl 21 | . 2 ⊢ (𝜑 → Rel (𝑋𝑇𝑌)) |
17 | relcnv 5961 | . . 3 ⊢ Rel ◡(𝑋𝑆𝑌) | |
18 | 17 | a1i 11 | . 2 ⊢ (𝜑 → Rel ◡(𝑋𝑆𝑌)) |
19 | oppcsect.s | . . . 4 ⊢ 𝑆 = (Sect‘𝐶) | |
20 | 2, 1, 8, 11, 12, 19, 7 | oppcsect 17255 | . . 3 ⊢ (𝜑 → (𝑓(𝑋𝑇𝑌)𝑔 ↔ 𝑔(𝑋𝑆𝑌)𝑓)) |
21 | vex 3405 | . . . 4 ⊢ 𝑓 ∈ V | |
22 | vex 3405 | . . . 4 ⊢ 𝑔 ∈ V | |
23 | 21, 22 | brcnv 5740 | . . 3 ⊢ (𝑓◡(𝑋𝑆𝑌)𝑔 ↔ 𝑔(𝑋𝑆𝑌)𝑓) |
24 | 20, 23 | bitr4di 292 | . 2 ⊢ (𝜑 → (𝑓(𝑋𝑇𝑌)𝑔 ↔ 𝑓◡(𝑋𝑆𝑌)𝑔)) |
25 | 16, 18, 24 | eqbrrdv 5652 | 1 ⊢ (𝜑 → (𝑋𝑇𝑌) = ◡(𝑋𝑆𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ⊆ wss 3857 class class class wbr 5043 × cxp 5538 ◡ccnv 5539 Rel wrel 5545 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 Hom chom 16778 compcco 16779 Catccat 17139 Idccid 17140 oppCatcoppc 17186 Sectcsect 17221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-hom 16791 df-cco 16792 df-cat 17143 df-cid 17144 df-oppc 17187 df-sect 17224 |
This theorem is referenced by: oppcinv 17257 |
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