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Theorem yonedalem3a 15670
Description: Lemma for yoneda 15679. (Contributed by Mario Carneiro, 29-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 )
yonedalem21.f  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
yonedalem21.x  |-  ( ph  ->  X  e.  B )
yonedalem3a.m  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
Assertion
Ref Expression
yonedalem3a  |-  ( ph  ->  ( ( F M X )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) )  /\  ( F M X ) : ( F ( 1st `  Z
) X ) --> ( F ( 1st `  E
) X ) ) )
Distinct variable groups:    f, a, x,  .1.    C, a, f, x    E, a, f    F, a, f, x    B, a, f, x    O, a, f, x    S, a, f, x    Q, a, f, x    T, f    ph, a, f, x    Y, a, f, x    Z, a, f, x    X, a, f, x
Allowed substitution hints:    R( x, f, a)    T( x, a)    U( x, f, a)    E( x)    H( x, f, a)    M( x, f, a)    V( x, f, a)    W( x, f, a)

Proof of Theorem yonedalem3a
StepHypRef Expression
1 yonedalem21.f . . 3  |-  ( ph  ->  F  e.  ( O 
Func  S ) )
2 yonedalem21.x . . 3  |-  ( ph  ->  X  e.  B )
3 simpr 461 . . . . . . 7  |-  ( ( f  =  F  /\  x  =  X )  ->  x  =  X )
43fveq2d 5876 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  X
) )
5 simpl 457 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  f  =  F )
64, 5oveq12d 6314 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ( ( 1st `  Y ) `  x
) ( O Nat  S
) f )  =  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )
73fveq2d 5876 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  ( a `  x
)  =  ( a `
 X ) )
83fveq2d 5876 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  (  .1.  `  x
)  =  (  .1.  `  X ) )
97, 8fveq12d 5878 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ( a `  x ) `  (  .1.  `  x ) )  =  ( ( a `
 X ) `  (  .1.  `  X )
) )
106, 9mpteq12dv 4535 . . . 4  |-  ( ( f  =  F  /\  x  =  X )  ->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) )  =  ( a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F )  |->  ( ( a `  X
) `  (  .1.  `  X ) ) ) )
11 yonedalem3a.m . . . 4  |-  M  =  ( f  e.  ( O  Func  S ) ,  x  e.  B  |->  ( a  e.  ( ( ( 1st `  Y
) `  x )
( O Nat  S ) f )  |->  ( ( a `  x ) `
 (  .1.  `  x ) ) ) )
12 ovex 6324 . . . . 5  |-  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  e.  _V
1312mptex 6144 . . . 4  |-  ( a  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( a `
 X ) `  (  .1.  `  X )
) )  e.  _V
1410, 11, 13ovmpt2a 6432 . . 3  |-  ( ( F  e.  ( O 
Func  S )  /\  X  e.  B )  ->  ( F M X )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) ) )
151, 2, 14syl2anc 661 . 2  |-  ( ph  ->  ( F M X )  =  ( a  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( a `
 X ) `  (  .1.  `  X )
) ) )
16 eqid 2457 . . . . . . 7  |-  ( O Nat 
S )  =  ( O Nat  S )
17 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F ) )
1816, 17nat1st2nd 15367 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  a  e.  (
<. ( 1st `  (
( 1st `  Y
) `  X )
) ,  ( 2nd `  ( ( 1st `  Y
) `  X )
) >. ( O Nat  S
) <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
)
19 yoneda.o . . . . . . . 8  |-  O  =  (oppCat `  C )
20 yoneda.b . . . . . . . 8  |-  B  =  ( Base `  C
)
2119, 20oppcbas 15134 . . . . . . 7  |-  B  =  ( Base `  O
)
22 eqid 2457 . . . . . . 7  |-  ( Hom  `  S )  =  ( Hom  `  S )
232adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  X  e.  B
)
2416, 18, 21, 22, 23natcl 15369 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  X )  e.  ( ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  X )
) )
25 yoneda.s . . . . . . 7  |-  S  =  ( SetCat `  U )
26 yoneda.w . . . . . . . . 9  |-  ( ph  ->  V  e.  W )
27 yoneda.v . . . . . . . . . 10  |-  ( ph  ->  ( ran  ( Hom f  `  Q )  u.  U
)  C_  V )
2827unssbd 3678 . . . . . . . . 9  |-  ( ph  ->  U  C_  V )
2926, 28ssexd 4603 . . . . . . . 8  |-  ( ph  ->  U  e.  _V )
3029adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  U  e.  _V )
31 eqid 2457 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
32 relfunc 15278 . . . . . . . . . . . 12  |-  Rel  ( O  Func  S )
33 yoneda.y . . . . . . . . . . . . 13  |-  Y  =  (Yon `  C )
34 yoneda.c . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  Cat )
35 yoneda.u . . . . . . . . . . . . 13  |-  ( ph  ->  ran  ( Hom f  `  C ) 
C_  U )
3633, 20, 34, 2, 19, 25, 29, 35yon1cl 15659 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
37 1st2ndbr 6848 . . . . . . . . . . . 12  |-  ( ( Rel  ( O  Func  S )  /\  ( ( 1st `  Y ) `
 X )  e.  ( O  Func  S
) )  ->  ( 1st `  ( ( 1st `  Y ) `  X
) ) ( O 
Func  S ) ( 2nd `  ( ( 1st `  Y
) `  X )
) )
3832, 36, 37sylancr 663 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) ( O  Func  S ) ( 2nd `  (
( 1st `  Y
) `  X )
) )
3921, 31, 38funcf1 15282 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> ( Base `  S ) )
4039, 2ffvelrnd 6033 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  e.  ( Base `  S
) )
4125, 29setcbas 15484 . . . . . . . . 9  |-  ( ph  ->  U  =  ( Base `  S ) )
4240, 41eleqtrrd 2548 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  e.  U )
4342adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  X )  e.  U )
44 1st2ndbr 6848 . . . . . . . . . . . 12  |-  ( ( Rel  ( O  Func  S )  /\  F  e.  ( O  Func  S
) )  ->  ( 1st `  F ) ( O  Func  S )
( 2nd `  F
) )
4532, 1, 44sylancr 663 . . . . . . . . . . 11  |-  ( ph  ->  ( 1st `  F
) ( O  Func  S ) ( 2nd `  F
) )
4621, 31, 45funcf1 15282 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  S ) )
4746, 2ffvelrnd 6033 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  ( Base `  S
) )
4847, 41eleqtrrd 2548 . . . . . . . 8  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  U )
4948adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( 1st `  F ) `  X
)  e.  U )
5025, 30, 22, 43, 49elsetchom 15487 . . . . . 6  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 X )  e.  ( ( ( 1st `  ( ( 1st `  Y
) `  X )
) `  X )
( Hom  `  S ) ( ( 1st `  F
) `  X )
)  <->  ( a `  X ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) --> ( ( 1st `  F
) `  X )
) )
5124, 50mpbid 210 . . . . 5  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( a `  X ) : ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X ) --> ( ( 1st `  F
) `  X )
)
52 eqid 2457 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
53 yoneda.1 . . . . . . . 8  |-  .1.  =  ( Id `  C )
5420, 52, 53, 34, 2catidcl 15099 . . . . . . 7  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X ( Hom  `  C
) X ) )
5533, 20, 34, 2, 52, 2yon11 15660 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  =  ( X ( Hom  `  C ) X ) )
5654, 55eleqtrrd 2548 . . . . . 6  |-  ( ph  ->  (  .1.  `  X
)  e.  ( ( 1st `  ( ( 1st `  Y ) `
 X ) ) `
 X ) )
5756adantr 465 . . . . 5  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  (  .1.  `  X )  e.  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )
)
5851, 57ffvelrnd 6033 . . . 4  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 X ) `  (  .1.  `  X )
)  e.  ( ( 1st `  F ) `
 X ) )
59 eqid 2457 . . . 4  |-  ( a  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( a `
 X ) `  (  .1.  `  X )
) )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) )
6058, 59fmptd 6056 . . 3  |-  ( ph  ->  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) ) : ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) --> ( ( 1st `  F ) `  X
) )
61 yoneda.t . . . . 5  |-  T  =  ( SetCat `  V )
62 yoneda.q . . . . 5  |-  Q  =  ( O FuncCat  S )
63 yoneda.h . . . . 5  |-  H  =  (HomF
`  Q )
64 yoneda.r . . . . 5  |-  R  =  ( ( Q  X.c  O
) FuncCat  T )
65 yoneda.e . . . . 5  |-  E  =  ( O evalF  S )
66 yoneda.z . . . . 5  |-  Z  =  ( H  o.func  ( ( <. ( 1st `  Y
) , tpos  ( 2nd `  Y ) >.  o.func  ( Q  2ndF  O ) ) ⟨,⟩F  ( Q  1stF  O )
) )
6733, 20, 53, 19, 25, 61, 62, 63, 64, 65, 66, 34, 26, 35, 27, 1, 2yonedalem21 15669 . . . 4  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
6819oppccat 15138 . . . . . 6  |-  ( C  e.  Cat  ->  O  e.  Cat )
6934, 68syl 16 . . . . 5  |-  ( ph  ->  O  e.  Cat )
7025setccat 15491 . . . . . 6  |-  ( U  e.  _V  ->  S  e.  Cat )
7129, 70syl 16 . . . . 5  |-  ( ph  ->  S  e.  Cat )
7265, 69, 71, 21, 1, 2evlf1 15616 . . . 4  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
7315, 67, 72feq123d 5727 . . 3  |-  ( ph  ->  ( ( F M X ) : ( F ( 1st `  Z
) X ) --> ( F ( 1st `  E
) X )  <->  ( a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F )  |->  ( ( a `  X
) `  (  .1.  `  X ) ) ) : ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) --> ( ( 1st `  F ) `  X
) ) )
7460, 73mpbird 232 . 2  |-  ( ph  ->  ( F M X ) : ( F ( 1st `  Z
) X ) --> ( F ( 1st `  E
) X ) )
7515, 74jca 532 1  |-  ( ph  ->  ( ( F M X )  =  ( a  e.  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  |->  ( ( a `  X ) `
 (  .1.  `  X ) ) )  /\  ( F M X ) : ( F ( 1st `  Z
) X ) --> ( F ( 1st `  E
) X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    C_ wss 3471   <.cop 4038   class class class wbr 4456    |-> cmpt 4515   ran crn 5009   Rel wrel 5013   -->wf 5590   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798  tpos ctpos 6972   Basecbs 14644   Hom chom 14723   Catccat 15081   Idccid 15082   Hom f chomf 15083  oppCatcoppc 15127    Func cfunc 15270    o.func ccofu 15272   Nat cnat 15357   FuncCat cfuc 15358   SetCatcsetc 15481    X.c cxpc 15564    1stF c1stf 15565    2ndF c2ndf 15566   ⟨,⟩F cprf 15567   evalF cevlf 15605  HomFchof 15644  Yoncyon 15645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-hom 14736  df-cco 14737  df-cat 15085  df-cid 15086  df-homf 15087  df-comf 15088  df-oppc 15128  df-func 15274  df-cofu 15276  df-nat 15359  df-fuc 15360  df-setc 15482  df-xpc 15568  df-1stf 15569  df-2ndf 15570  df-prf 15571  df-evlf 15609  df-curf 15610  df-hof 15646  df-yon 15647
This theorem is referenced by:  yonedalem3b  15675  yonedalem3  15676  yonedainv  15677
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