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Mirrors > Home > MPE Home > Th. List > yonedalem1 | Structured version Visualization version GIF version |
Description: Lemma for yoneda 18177. (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 2733 | . . . . 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 2733 | . . . . 5 β’ ((oppCatβπ) Γc π) = ((oppCatβπ) Γc π) | |
4 | eqid 2733 | . . . . . . 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 17609 | . . . . . . . . 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 4149 | . . . . . . . . . 10 β’ (π β π β π) |
13 | 10, 12 | ssexd 5282 | . . . . . . . . 9 β’ (π β π β V) |
14 | yoneda.s | . . . . . . . . . 10 β’ π = (SetCatβπ) | |
15 | 14 | setccat 17976 | . . . . . . . . 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 17864 | . . . . . . 7 β’ (π β π β Cat) |
18 | eqid 2733 | . . . . . . 7 β’ (π 2ndF π) = (π 2ndF π) | |
19 | 4, 17, 9, 18 | 2ndfcl 18091 | . . . . . 6 β’ (π β (π 2ndF π) β ((π Γc π) Func π)) |
20 | eqid 2733 | . . . . . . . 8 β’ (oppCatβπ) = (oppCatβπ) | |
21 | relfunc 17753 | . . . . . . . . 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 18156 | . . . . . . . . 9 β’ (π β π β (πΆ Func π)) |
25 | 1st2ndbr 7975 | . . . . . . . . 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 588 | . . . . . . . 8 β’ (π β (1st βπ)(πΆ Func π)(2nd βπ)) |
27 | 7, 20, 26 | funcoppc 17766 | . . . . . . 7 β’ (π β (1st βπ)(π Func (oppCatβπ))tpos (2nd βπ)) |
28 | df-br 5107 | . . . . . . 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 17779 | . . . . 5 β’ (π β (β¨(1st βπ), tpos (2nd βπ)β© βfunc (π 2ndF π)) β ((π Γc π) Func (oppCatβπ))) |
31 | eqid 2733 | . . . . . 6 β’ (π 1stF π) = (π 1stF π) | |
32 | 4, 17, 9, 31 | 1stfcl 18090 | . . . . 5 β’ (π β (π 1stF π) β ((π Γc π) Func π)) |
33 | 2, 3, 30, 32 | prfcl 18096 | . . . 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 4148 | . . . . 5 β’ (π β ran (Homf βπ) β π) |
37 | 34, 20, 35, 17, 10, 36 | hofcl 18153 | . . . 4 β’ (π β π» β (((oppCatβπ) Γc π) Func π)) |
38 | 33, 37 | cofucl 17779 | . . 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 2838 | . 2 β’ (π β π β ((π Γc π) Func π)) |
40 | 35, 14, 10, 12 | funcsetcres2 17984 | . . 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 18116 | . . 3 β’ (π β πΈ β ((π Γc π) Func π)) |
43 | 40, 42 | sseldd 3946 | . 2 β’ (π β πΈ β ((π Γc π) Func π)) |
44 | 39, 43 | jca 513 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3444 βͺ cun 3909 β wss 3911 β¨cop 4593 class class class wbr 5106 ran crn 5635 Rel wrel 5639 βcfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 tpos ctpos 8157 Basecbs 17088 Catccat 17549 Idccid 17550 Homf chomf 17551 oppCatcoppc 17596 Func cfunc 17745 βfunc ccofu 17747 FuncCat cfuc 17834 SetCatcsetc 17966 Γc cxpc 18061 1stF c1stf 18062 2ndF c2ndf 18063 β¨,β©F cprf 18064 evalF cevlf 18103 HomFchof 18142 Yoncyon 18143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-map 8770 df-pm 8771 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-hom 17162 df-cco 17163 df-cat 17553 df-cid 17554 df-homf 17555 df-comf 17556 df-oppc 17597 df-ssc 17698 df-resc 17699 df-subc 17700 df-func 17749 df-cofu 17751 df-nat 17835 df-fuc 17836 df-setc 17967 df-xpc 18065 df-1stf 18066 df-2ndf 18067 df-prf 18068 df-evlf 18107 df-curf 18108 df-hof 18144 df-yon 18145 |
This theorem is referenced by: yonedalem3b 18173 yonedalem3 18174 yonedainv 18175 yonffthlem 18176 yoneda 18177 |
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