![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > yonedalem1 | Structured version Visualization version GIF version |
Description: Lemma for yoneda 17309. (Contributed by Mario Carneiro, 28-Jan-2017.) |
Ref | Expression |
---|---|
yoneda.y | ⊢ 𝑌 = (Yon‘𝐶) |
yoneda.b | ⊢ 𝐵 = (Base‘𝐶) |
yoneda.1 | ⊢ 1 = (Id‘𝐶) |
yoneda.o | ⊢ 𝑂 = (oppCat‘𝐶) |
yoneda.s | ⊢ 𝑆 = (SetCat‘𝑈) |
yoneda.t | ⊢ 𝑇 = (SetCat‘𝑉) |
yoneda.q | ⊢ 𝑄 = (𝑂 FuncCat 𝑆) |
yoneda.h | ⊢ 𝐻 = (HomF‘𝑄) |
yoneda.r | ⊢ 𝑅 = ((𝑄 ×c 𝑂) FuncCat 𝑇) |
yoneda.e | ⊢ 𝐸 = (𝑂 evalF 𝑆) |
yoneda.z | ⊢ 𝑍 = (𝐻 ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) |
yoneda.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
yoneda.w | ⊢ (𝜑 → 𝑉 ∈ 𝑊) |
yoneda.u | ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) |
yoneda.v | ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) |
Ref | Expression |
---|---|
yonedalem1 | ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yoneda.z | . . 3 ⊢ 𝑍 = (𝐻 ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) | |
2 | eqid 2777 | . . . . 5 ⊢ ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂)) = ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂)) | |
3 | eqid 2777 | . . . . 5 ⊢ ((oppCat‘𝑄) ×c 𝑄) = ((oppCat‘𝑄) ×c 𝑄) | |
4 | eqid 2777 | . . . . . . 7 ⊢ (𝑄 ×c 𝑂) = (𝑄 ×c 𝑂) | |
5 | yoneda.q | . . . . . . . 8 ⊢ 𝑄 = (𝑂 FuncCat 𝑆) | |
6 | yoneda.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
7 | yoneda.o | . . . . . . . . . 10 ⊢ 𝑂 = (oppCat‘𝐶) | |
8 | 7 | oppccat 16767 | . . . . . . . . 9 ⊢ (𝐶 ∈ Cat → 𝑂 ∈ Cat) |
9 | 6, 8 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ Cat) |
10 | yoneda.w | . . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ 𝑊) | |
11 | yoneda.v | . . . . . . . . . . 11 ⊢ (𝜑 → (ran (Homf ‘𝑄) ∪ 𝑈) ⊆ 𝑉) | |
12 | 11 | unssbd 4013 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑉) |
13 | 10, 12 | ssexd 5042 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ V) |
14 | yoneda.s | . . . . . . . . . 10 ⊢ 𝑆 = (SetCat‘𝑈) | |
15 | 14 | setccat 17120 | . . . . . . . . 9 ⊢ (𝑈 ∈ V → 𝑆 ∈ Cat) |
16 | 13, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Cat) |
17 | 5, 9, 16 | fuccat 17015 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ Cat) |
18 | eqid 2777 | . . . . . . 7 ⊢ (𝑄 2ndF 𝑂) = (𝑄 2ndF 𝑂) | |
19 | 4, 17, 9, 18 | 2ndfcl 17224 | . . . . . 6 ⊢ (𝜑 → (𝑄 2ndF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑂)) |
20 | eqid 2777 | . . . . . . . 8 ⊢ (oppCat‘𝑄) = (oppCat‘𝑄) | |
21 | relfunc 16907 | . . . . . . . . 9 ⊢ Rel (𝐶 Func 𝑄) | |
22 | yoneda.y | . . . . . . . . . 10 ⊢ 𝑌 = (Yon‘𝐶) | |
23 | yoneda.u | . . . . . . . . . 10 ⊢ (𝜑 → ran (Homf ‘𝐶) ⊆ 𝑈) | |
24 | 22, 6, 7, 14, 5, 13, 23 | yoncl 17288 | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝐶 Func 𝑄)) |
25 | 1st2ndbr 7496 | . . . . . . . . 9 ⊢ ((Rel (𝐶 Func 𝑄) ∧ 𝑌 ∈ (𝐶 Func 𝑄)) → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) | |
26 | 21, 24, 25 | sylancr 581 | . . . . . . . 8 ⊢ (𝜑 → (1st ‘𝑌)(𝐶 Func 𝑄)(2nd ‘𝑌)) |
27 | 7, 20, 26 | funcoppc 16920 | . . . . . . 7 ⊢ (𝜑 → (1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌)) |
28 | df-br 4887 | . . . . . . 7 ⊢ ((1st ‘𝑌)(𝑂 Func (oppCat‘𝑄))tpos (2nd ‘𝑌) ↔ 〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∈ (𝑂 Func (oppCat‘𝑄))) | |
29 | 27, 28 | sylib 210 | . . . . . 6 ⊢ (𝜑 → 〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∈ (𝑂 Func (oppCat‘𝑄))) |
30 | 19, 29 | cofucl 16933 | . . . . 5 ⊢ (𝜑 → (〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func (oppCat‘𝑄))) |
31 | eqid 2777 | . . . . . 6 ⊢ (𝑄 1stF 𝑂) = (𝑄 1stF 𝑂) | |
32 | 4, 17, 9, 31 | 1stfcl 17223 | . . . . 5 ⊢ (𝜑 → (𝑄 1stF 𝑂) ∈ ((𝑄 ×c 𝑂) Func 𝑄)) |
33 | 2, 3, 30, 32 | prfcl 17229 | . . . 4 ⊢ (𝜑 → ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂)) ∈ ((𝑄 ×c 𝑂) Func ((oppCat‘𝑄) ×c 𝑄))) |
34 | yoneda.h | . . . . 5 ⊢ 𝐻 = (HomF‘𝑄) | |
35 | yoneda.t | . . . . 5 ⊢ 𝑇 = (SetCat‘𝑉) | |
36 | 11 | unssad 4012 | . . . . 5 ⊢ (𝜑 → ran (Homf ‘𝑄) ⊆ 𝑉) |
37 | 34, 20, 35, 17, 10, 36 | hofcl 17285 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ (((oppCat‘𝑄) ×c 𝑄) Func 𝑇)) |
38 | 33, 37 | cofucl 16933 | . . 3 ⊢ (𝜑 → (𝐻 ∘func ((〈(1st ‘𝑌), tpos (2nd ‘𝑌)〉 ∘func (𝑄 2ndF 𝑂)) 〈,〉F (𝑄 1stF 𝑂))) ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
39 | 1, 38 | syl5eqel 2862 | . 2 ⊢ (𝜑 → 𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
40 | 35, 14, 10, 12 | funcsetcres2 17128 | . . 3 ⊢ (𝜑 → ((𝑄 ×c 𝑂) Func 𝑆) ⊆ ((𝑄 ×c 𝑂) Func 𝑇)) |
41 | yoneda.e | . . . 4 ⊢ 𝐸 = (𝑂 evalF 𝑆) | |
42 | 41, 5, 9, 16 | evlfcl 17248 | . . 3 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑆)) |
43 | 40, 42 | sseldd 3821 | . 2 ⊢ (𝜑 → 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇)) |
44 | 39, 43 | jca 507 | 1 ⊢ (𝜑 → (𝑍 ∈ ((𝑄 ×c 𝑂) Func 𝑇) ∧ 𝐸 ∈ ((𝑄 ×c 𝑂) Func 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ∪ cun 3789 ⊆ wss 3791 〈cop 4403 class class class wbr 4886 ran crn 5356 Rel wrel 5360 ‘cfv 6135 (class class class)co 6922 1st c1st 7443 2nd c2nd 7444 tpos ctpos 7633 Basecbs 16255 Catccat 16710 Idccid 16711 Homf chomf 16712 oppCatcoppc 16756 Func cfunc 16899 ∘func ccofu 16901 FuncCat cfuc 16987 SetCatcsetc 17110 ×c cxpc 17194 1stF c1stf 17195 2ndF c2ndf 17196 〈,〉F cprf 17197 evalF cevlf 17235 HomFchof 17274 Yoncyon 17275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-tpos 7634 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-fz 12644 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-hom 16362 df-cco 16363 df-cat 16714 df-cid 16715 df-homf 16716 df-comf 16717 df-oppc 16757 df-ssc 16855 df-resc 16856 df-subc 16857 df-func 16903 df-cofu 16905 df-nat 16988 df-fuc 16989 df-setc 17111 df-xpc 17198 df-1stf 17199 df-2ndf 17200 df-prf 17201 df-evlf 17239 df-curf 17240 df-hof 17276 df-yon 17277 |
This theorem is referenced by: yonedalem3b 17305 yonedalem3 17306 yonedainv 17307 yonffthlem 17308 yoneda 17309 |
Copyright terms: Public domain | W3C validator |