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Theorem yonedalem1 15186
Description: Lemma for yoneda 15197. (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 2451 . . . . 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 2451 . . . . 5  |-  ( (oppCat `  Q )  X.c  Q )  =  ( (oppCat `  Q )  X.c  Q )
4 eqid 2451 . . . . . . 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 14765 . . . . . . . . 9  |-  ( C  e.  Cat  ->  O  e.  Cat )
96, 8syl 16 . . . . . . . 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 3634 . . . . . . . . . 10  |-  ( ph  ->  U  C_  V )
1310, 12ssexd 4539 . . . . . . . . 9  |-  ( ph  ->  U  e.  _V )
14 yoneda.s . . . . . . . . . 10  |-  S  =  ( SetCat `  U )
1514setccat 15057 . . . . . . . . 9  |-  ( U  e.  _V  ->  S  e.  Cat )
1613, 15syl 16 . . . . . . . 8  |-  ( ph  ->  S  e.  Cat )
175, 9, 16fuccat 14984 . . . . . . 7  |-  ( ph  ->  Q  e.  Cat )
18 eqid 2451 . . . . . . 7  |-  ( Q  2ndF  O )  =  ( Q  2ndF  O )
194, 17, 9, 182ndfcl 15112 . . . . . 6  |-  ( ph  ->  ( Q  2ndF  O )  e.  ( ( Q  X.c  O
)  Func  O )
)
20 eqid 2451 . . . . . . . 8  |-  (oppCat `  Q )  =  (oppCat `  Q )
21 relfunc 14876 . . . . . . . . 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 15176 . . . . . . . . 9  |-  ( ph  ->  Y  e.  ( C 
Func  Q ) )
25 1st2ndbr 6725 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  Q )  /\  Y  e.  ( C  Func  Q
) )  ->  ( 1st `  Y ) ( C  Func  Q )
( 2nd `  Y
) )
2621, 24, 25sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 1st `  Y
) ( C  Func  Q ) ( 2nd `  Y
) )
277, 20, 26funcoppc 14889 . . . . . . 7  |-  ( ph  ->  ( 1st `  Y
) ( O  Func  (oppCat `  Q ) )tpos  ( 2nd `  Y ) )
28 df-br 4393 . . . . . . 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 14902 . . . . 5  |-  ( ph  ->  ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) )  e.  ( ( Q  X.c  O ) 
Func  (oppCat `  Q )
) )
31 eqid 2451 . . . . . 6  |-  ( Q  1stF  O )  =  ( Q  1stF  O )
324, 17, 9, 311stfcl 15111 . . . . 5  |-  ( ph  ->  ( Q  1stF  O )  e.  ( ( Q  X.c  O
)  Func  Q )
)
332, 3, 30, 32prfcl 15117 . . . 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 3633 . . . . 5  |-  ( ph  ->  ran  ( Hom f  `  Q ) 
C_  V )
3734, 20, 35, 17, 10, 36hofcl 15173 . . . 4  |-  ( ph  ->  H  e.  ( ( (oppCat `  Q )  X.c  Q )  Func  T
) )
3833, 37cofucl 14902 . . 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 2543 . 2  |-  ( ph  ->  Z  e.  ( ( Q  X.c  O )  Func  T
) )
4035, 14, 10, 12funcsetcres2 15065 . . 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 15136 . . 3  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  S
) )
4340, 42sseldd 3457 . 2  |-  ( ph  ->  E  e.  ( ( Q  X.c  O )  Func  T
) )
4439, 43jca 532 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 369    = wceq 1370    e. wcel 1758   _Vcvv 3070    u. cun 3426    C_ wss 3428   <.cop 3983   class class class wbr 4392   ran crn 4941   Rel wrel 4945   ` cfv 5518  (class class class)co 6192   1stc1st 6677   2ndc2nd 6678  tpos ctpos 6846   Basecbs 14278   Catccat 14706   Idccid 14707   Hom f chomf 14708  oppCatcoppc 14754    Func cfunc 14868    o.func ccofu 14870   FuncCat cfuc 14956   SetCatcsetc 15047    X.c cxpc 15082    1stF c1stf 15083    2ndF c2ndf 15084   ⟨,⟩F cprf 15085   evalF cevlf 15123  HomFchof 15162  Yoncyon 15163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-tpos 6847  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-ixp 7366  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-hom 14366  df-cco 14367  df-cat 14710  df-cid 14711  df-homf 14712  df-comf 14713  df-oppc 14755  df-ssc 14827  df-resc 14828  df-subc 14829  df-func 14872  df-cofu 14874  df-nat 14957  df-fuc 14958  df-setc 15048  df-xpc 15086  df-1stf 15087  df-2ndf 15088  df-prf 15089  df-evlf 15127  df-curf 15128  df-hof 15164  df-yon 15165
This theorem is referenced by:  yonedalem3b  15193  yonedalem3  15194  yonedainv  15195  yonffthlem  15196  yoneda  15197
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