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Mirrors > Home > MPE Home > Th. List > yonedalem1 | Structured version Visualization version GIF version |
Description: Lemma for yoneda 18272. (Contributed by Mario Carneiro, 28-Jan-2017.) |
Ref | Expression |
---|---|
yoneda.y | β’ π = (YonβπΆ) |
yoneda.b | β’ π΅ = (BaseβπΆ) |
yoneda.1 | β’ 1 = (IdβπΆ) |
yoneda.o | β’ π = (oppCatβπΆ) |
yoneda.s | β’ π = (SetCatβπ) |
yoneda.t | β’ π = (SetCatβπ) |
yoneda.q | β’ π = (π FuncCat π) |
yoneda.h | β’ π» = (HomFβπ) |
yoneda.r | β’ π = ((π Γc π) FuncCat π) |
yoneda.e | β’ πΈ = (π evalF π) |
yoneda.z | β’ π = (π» βfunc ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π))) |
yoneda.c | β’ (π β πΆ β Cat) |
yoneda.w | β’ (π β π β π) |
yoneda.u | β’ (π β ran (Homf βπΆ) β π) |
yoneda.v | β’ (π β (ran (Homf βπ) βͺ π) β π) |
Ref | Expression |
---|---|
yonedalem1 | β’ (π β (π β ((π Γc π) Func π) β§ πΈ β ((π Γc π) Func π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | yoneda.z | . . 3 β’ π = (π» βfunc ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π))) | |
2 | eqid 2725 | . . . . 5 β’ ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π)) = ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π)) | |
3 | eqid 2725 | . . . . 5 β’ ((oppCatβπ) Γc π) = ((oppCatβπ) Γc π) | |
4 | eqid 2725 | . . . . . . 7 β’ (π Γc π) = (π Γc π) | |
5 | yoneda.q | . . . . . . . 8 β’ π = (π FuncCat π) | |
6 | yoneda.c | . . . . . . . . 9 β’ (π β πΆ β Cat) | |
7 | yoneda.o | . . . . . . . . . 10 β’ π = (oppCatβπΆ) | |
8 | 7 | oppccat 17701 | . . . . . . . . 9 β’ (πΆ β Cat β π β Cat) |
9 | 6, 8 | syl 17 | . . . . . . . 8 β’ (π β π β Cat) |
10 | yoneda.w | . . . . . . . . . 10 β’ (π β π β π) | |
11 | yoneda.v | . . . . . . . . . . 11 β’ (π β (ran (Homf βπ) βͺ π) β π) | |
12 | 11 | unssbd 4182 | . . . . . . . . . 10 β’ (π β π β π) |
13 | 10, 12 | ssexd 5319 | . . . . . . . . 9 β’ (π β π β V) |
14 | yoneda.s | . . . . . . . . . 10 β’ π = (SetCatβπ) | |
15 | 14 | setccat 18071 | . . . . . . . . 9 β’ (π β V β π β Cat) |
16 | 13, 15 | syl 17 | . . . . . . . 8 β’ (π β π β Cat) |
17 | 5, 9, 16 | fuccat 17959 | . . . . . . 7 β’ (π β π β Cat) |
18 | eqid 2725 | . . . . . . 7 β’ (π 2ndF π) = (π 2ndF π) | |
19 | 4, 17, 9, 18 | 2ndfcl 18186 | . . . . . 6 β’ (π β (π 2ndF π) β ((π Γc π) Func π)) |
20 | eqid 2725 | . . . . . . . 8 β’ (oppCatβπ) = (oppCatβπ) | |
21 | relfunc 17845 | . . . . . . . . 9 β’ Rel (πΆ Func π) | |
22 | yoneda.y | . . . . . . . . . 10 β’ π = (YonβπΆ) | |
23 | yoneda.u | . . . . . . . . . 10 β’ (π β ran (Homf βπΆ) β π) | |
24 | 22, 6, 7, 14, 5, 13, 23 | yoncl 18251 | . . . . . . . . 9 β’ (π β π β (πΆ Func π)) |
25 | 1st2ndbr 8042 | . . . . . . . . 9 β’ ((Rel (πΆ Func π) β§ π β (πΆ Func π)) β (1st βπ)(πΆ Func π)(2nd βπ)) | |
26 | 21, 24, 25 | sylancr 585 | . . . . . . . 8 β’ (π β (1st βπ)(πΆ Func π)(2nd βπ)) |
27 | 7, 20, 26 | funcoppc 17858 | . . . . . . 7 β’ (π β (1st βπ)(π Func (oppCatβπ))tpos (2nd βπ)) |
28 | df-br 5144 | . . . . . . 7 β’ ((1st βπ)(π Func (oppCatβπ))tpos (2nd βπ) β β¨(1st βπ), tpos (2nd βπ)β© β (π Func (oppCatβπ))) | |
29 | 27, 28 | sylib 217 | . . . . . 6 β’ (π β β¨(1st βπ), tpos (2nd βπ)β© β (π Func (oppCatβπ))) |
30 | 19, 29 | cofucl 17871 | . . . . 5 β’ (π β (β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β ((π Γc π) Func (oppCatβπ))) |
31 | eqid 2725 | . . . . . 6 β’ (π 1stF π) = (π 1stF π) | |
32 | 4, 17, 9, 31 | 1stfcl 18185 | . . . . 5 β’ (π β (π 1stF π) β ((π Γc π) Func π)) |
33 | 2, 3, 30, 32 | prfcl 18191 | . . . 4 β’ (π β ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π)) β ((π Γc π) Func ((oppCatβπ) Γc π))) |
34 | yoneda.h | . . . . 5 β’ π» = (HomFβπ) | |
35 | yoneda.t | . . . . 5 β’ π = (SetCatβπ) | |
36 | 11 | unssad 4181 | . . . . 5 β’ (π β ran (Homf βπ) β π) |
37 | 34, 20, 35, 17, 10, 36 | hofcl 18248 | . . . 4 β’ (π β π» β (((oppCatβπ) Γc π) Func π)) |
38 | 33, 37 | cofucl 17871 | . . 3 β’ (π β (π» βfunc ((β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β¨,β©F (π 1stF π))) β ((π Γc π) Func π)) |
39 | 1, 38 | eqeltrid 2829 | . 2 β’ (π β π β ((π Γc π) Func π)) |
40 | 35, 14, 10, 12 | funcsetcres2 18079 | . . 3 β’ (π β ((π Γc π) Func π) β ((π Γc π) Func π)) |
41 | yoneda.e | . . . 4 β’ πΈ = (π evalF π) | |
42 | 41, 5, 9, 16 | evlfcl 18211 | . . 3 β’ (π β πΈ β ((π Γc π) Func π)) |
43 | 40, 42 | sseldd 3973 | . 2 β’ (π β πΈ β ((π Γc π) Func π)) |
44 | 39, 43 | jca 510 | 1 β’ (π β (π β ((π Γc π) Func π) β§ πΈ β ((π Γc π) Func π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 Vcvv 3463 βͺ cun 3938 β wss 3940 β¨cop 4630 class class class wbr 5143 ran crn 5673 Rel wrel 5677 βcfv 6542 (class class class)co 7415 1st c1st 7987 2nd c2nd 7988 tpos ctpos 8227 Basecbs 17177 Catccat 17641 Idccid 17642 Homf chomf 17643 oppCatcoppc 17688 Func cfunc 17837 βfunc ccofu 17839 FuncCat cfuc 17929 SetCatcsetc 18061 Γc cxpc 18156 1stF c1stf 18157 2ndF c2ndf 18158 β¨,β©F cprf 18159 evalF cevlf 18198 HomFchof 18237 Yoncyon 18238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-pm 8844 df-ixp 8913 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-hom 17254 df-cco 17255 df-cat 17645 df-cid 17646 df-homf 17647 df-comf 17648 df-oppc 17689 df-ssc 17790 df-resc 17791 df-subc 17792 df-func 17841 df-cofu 17843 df-nat 17930 df-fuc 17931 df-setc 18062 df-xpc 18160 df-1stf 18161 df-2ndf 18162 df-prf 18163 df-evlf 18202 df-curf 18203 df-hof 18239 df-yon 18240 |
This theorem is referenced by: yonedalem3b 18268 yonedalem3 18269 yonedainv 18270 yonffthlem 18271 yoneda 18272 |
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