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Theorem yonedalem1 15865
Description: Lemma for yoneda 15876. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
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  X.c  O
) FuncCat  T )
yoneda.e  |-  E  =  ( O evalF  S )
yoneda.z  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
yoneda.c  |-  ( ph  ->  C  e.  Cat )
yoneda.w  |-  ( ph  ->  V  e.  W )
yoneda.u  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
yoneda.v  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
Assertion
Ref Expression
yonedalem1  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )

Proof of Theorem yonedalem1
StepHypRef Expression
1 yoneda.z . . 3  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
2 eqid 2402 . . . . 5  |-  ( (
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
)  =  ( (
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
)
3 eqid 2402 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
4 eqid 2402 . . . . . . 7  |-  ( Q  X.c  O )  =  ( Q  X.c  O )
5 yoneda.q . . . . . . . 8  |-  Q  =  ( O FuncCat  S )
6 yoneda.c . . . . . . . . 9  |-  ( ph  ->  C  e.  Cat )
7 yoneda.o . . . . . . . . . 10  |-  O  =  (oppCat `  C )
87oppccat 15335 . . . . . . . . 9  |-  ( C  e.  Cat  ->  O  e.  Cat )
96, 8syl 17 . . . . . . . 8  |-  ( ph  ->  O  e.  Cat )
10 yoneda.w . . . . . . . . . 10  |-  ( ph  ->  V  e.  W )
11 yoneda.v . . . . . . . . . . 11  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
1211unssbd 3621 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
1310, 12ssexd 4541 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
14 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
1514setccat 15688 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
1613, 15syl 17 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
175, 9, 16fuccat 15583 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
18 eqid 2402 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
194, 17, 9, 182ndfcl 15791 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
20 eqid 2402 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
21 relfunc 15475 . . . . . . . . 9  |-  Rel  ( C  Func  Q )
22 yoneda.y . . . . . . . . . 10  |-  Y  =  (Yon `  C )
23 yoneda.u . . . . . . . . . 10  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
2422, 6, 7, 14, 5, 13, 23yoncl 15855 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
25 1st2ndbr 6833 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
2621, 24, 25sylancr 661 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
277, 20, 26funcoppc 15488 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
28 df-br 4396 . . . . . . 7  |-  ( ( 1st `  Y ) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y )  <->  <. ( 1st `  Y ) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q ) ) )
2927, 28sylib 196 . . . . . 6  |-  ( ph  -> 
<. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  e.  ( O  Func  (oppCat `  Q
) ) )
3019, 29cofucl 15501 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
31 eqid 2402 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
324, 17, 9, 311stfcl 15790 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
332, 3, 30, 32prfcl 15796 . . . 4  |-  ( ph  ->  ( ( <. ( 1st `  Y ) , tpos  ( 2nd `  Y
) >.  o.func  ( Q  2ndF  O )
) ⟨,⟩F  ( Q  1stF  O ) )  e.  ( ( Q  X.c  O
)  Func  ( (oppCat `  Q )  X.c  Q ) ) )
34 yoneda.h . . . . 5  |-  H  =  (HomF
`  Q )
35 yoneda.t . . . . 5  |-  T  =  ( SetCat `  V )
3611unssad 3620 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
3734, 20, 35, 17, 10, 36hofcl 15852 . . . 4  |-  ( ph  ->  H  e.  ( ( (oppCat `  Q )  X.c  Q )  Func  T
) )
3833, 37cofucl 15501 . . 3  |-  ( ph  ->  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )  e.  ( ( Q  X.c  O ) 
Func  T ) )
391, 38syl5eqel 2494 . 2  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
4035, 14, 10, 12funcsetcres2 15696 . . 3  |-  ( ph  ->  ( ( Q  X.c  O
)  Func  S )  C_  ( ( Q  X.c  O
)  Func  T )
)
41 yoneda.e . . . 4  |-  E  =  ( O evalF  S )
4241, 5, 9, 16evlfcl 15815 . . 3  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  S
) )
4340, 42sseldd 3443 . 2  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
4439, 43jca 530 1  |-  ( ph  ->  ( Z  e.  ( ( Q  X.c  O ) 
Func  T )  /\  E  e.  ( ( Q  X.c  O
)  Func  T )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   _Vcvv 3059    u. cun 3412    C_ wss 3414   <.cop 3978   class class class wbr 4395   ran crn 4824   Rel wrel 4828   ` cfv 5569  (class class class)co 6278   1stc1st 6782   2ndc2nd 6783  tpos ctpos 6957   Basecbs 14841   Catccat 15278   Idccid 15279   Hom f chomf 15280  oppCatcoppc 15324    Func cfunc 15467    o.func ccofu 15469   FuncCat cfuc 15555   SetCatcsetc 15678    X.c cxpc 15761    1stF c1stf 15762    2ndF c2ndf 15763   ⟨,⟩F cprf 15764   evalF cevlf 15802  HomFchof 15841  Yoncyon 15842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-hom 14933  df-cco 14934  df-cat 15282  df-cid 15283  df-homf 15284  df-comf 15285  df-oppc 15325  df-ssc 15423  df-resc 15424  df-subc 15425  df-func 15471  df-cofu 15473  df-nat 15556  df-fuc 15557  df-setc 15679  df-xpc 15765  df-1stf 15766  df-2ndf 15767  df-prf 15768  df-evlf 15806  df-curf 15807  df-hof 15843  df-yon 15844
This theorem is referenced by:  yonedalem3b  15872  yonedalem3  15873  yonedainv  15874  yonffthlem  15875  yoneda  15876
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