![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 16859 | . . . 4 ⊢ 𝐵 = (Base‘𝑂) |
4 | eqid 2773 | . . . 4 ⊢ (Hom ‘𝑂) = (Hom ‘𝑂) | |
5 | eqid 2773 | . . . 4 ⊢ (comp‘𝑂) = (comp‘𝑂) | |
6 | eqid 2773 | . . . 4 ⊢ (Id‘𝑂) = (Id‘𝑂) | |
7 | oppcsect.t | . . . 4 ⊢ 𝑇 = (Sect‘𝑂) | |
8 | oppcsect.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
9 | 1 | oppccat 16863 | . . . . 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 16893 | . . 3 ⊢ (𝜑 → (𝑋𝑇𝑌) ⊆ ((𝑋(Hom ‘𝑂)𝑌) × (𝑌(Hom ‘𝑂)𝑋))) |
14 | relxp 5422 | . . 3 ⊢ Rel ((𝑋(Hom ‘𝑂)𝑌) × (𝑌(Hom ‘𝑂)𝑋)) | |
15 | relss 5503 | . . 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 5805 | . . 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 16919 | . . 3 ⊢ (𝜑 → (𝑓(𝑋𝑇𝑌)𝑔 ↔ 𝑔(𝑋𝑆𝑌)𝑓)) |
21 | vex 3413 | . . . 4 ⊢ 𝑓 ∈ V | |
22 | vex 3413 | . . . 4 ⊢ 𝑔 ∈ V | |
23 | 21, 22 | brcnv 5600 | . . 3 ⊢ (𝑓◡(𝑋𝑆𝑌)𝑔 ↔ 𝑔(𝑋𝑆𝑌)𝑓) |
24 | 20, 23 | syl6bbr 281 | . 2 ⊢ (𝜑 → (𝑓(𝑋𝑇𝑌)𝑔 ↔ 𝑓◡(𝑋𝑆𝑌)𝑔)) |
25 | 16, 18, 24 | eqbrrdv 5513 | 1 ⊢ (𝜑 → (𝑋𝑇𝑌) = ◡(𝑋𝑆𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 ⊆ wss 3824 class class class wbr 4926 × cxp 5402 ◡ccnv 5403 Rel wrel 5409 ‘cfv 6186 (class class class)co 6975 Basecbs 16338 Hom chom 16431 compcco 16432 Catccat 16806 Idccid 16807 oppCatcoppc 16852 Sectcsect 16885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-1st 7500 df-2nd 7501 df-tpos 7694 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-5 11505 df-6 11506 df-7 11507 df-8 11508 df-9 11509 df-n0 11707 df-z 11793 df-dec 11911 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-hom 16444 df-cco 16445 df-cat 16810 df-cid 16811 df-oppc 16853 df-sect 16888 |
This theorem is referenced by: oppcinv 16921 |
Copyright terms: Public domain | W3C validator |