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Theorem yonedalem3a 15105
Description: Lemma for yoneda 15114. (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 5716 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ( 1st `  Y
) `  x )  =  ( ( 1st `  Y ) `  X
) )
5 simpl 457 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  f  =  F )
64, 5oveq12d 6130 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ( ( 1st `  Y ) `  x
) ( O Nat  S
) f )  =  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )
73fveq2d 5716 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  ( a `  x
)  =  ( a `
 X ) )
83fveq2d 5716 . . . . . 6  |-  ( ( f  =  F  /\  x  =  X )  ->  (  .1.  `  x
)  =  (  .1.  `  X ) )
97, 8fveq12d 5718 . . . . 5  |-  ( ( f  =  F  /\  x  =  X )  ->  ( ( a `  x ) `  (  .1.  `  x ) )  =  ( ( a `
 X ) `  (  .1.  `  X )
) )
106, 9mpteq12dv 4391 . . . 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 6137 . . . . 5  |-  ( ( ( 1st `  Y
) `  X )
( O Nat  S ) F )  e.  _V
1312mptex 5969 . . . 4  |-  ( a  e.  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F )  |->  ( ( a `
 X ) `  (  .1.  `  X )
) )  e.  _V
1410, 11, 13ovmpt2a 6242 . . 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 2443 . . . . . . 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 14882 . . . . . . 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 14678 . . . . . . 7  |-  B  =  ( Base `  O
)
22 eqid 2443 . . . . . . 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 14884 . . . . . 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 3555 . . . . . . . . 9  |-  ( ph  ->  U  C_  V )
2926, 28ssexd 4460 . . . . . . . 8  |-  ( ph  ->  U  e.  _V )
3029adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  U  e.  _V )
31 eqid 2443 . . . . . . . . . . 11  |-  ( Base `  S )  =  (
Base `  S )
32 relfunc 14793 . . . . . . . . . . . 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 15094 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 1st `  Y
) `  X )  e.  ( O  Func  S
) )
37 1st2ndbr 6644 . . . . . . . . . . . 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 14797 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  (
( 1st `  Y
) `  X )
) : B --> ( Base `  S ) )
4039, 2ffvelrnd 5865 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  e.  ( Base `  S
) )
4125, 29setcbas 14967 . . . . . . . . 9  |-  ( ph  ->  U  =  ( Base `  S ) )
4240, 41eleqtrrd 2520 . . . . . . . 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 6644 . . . . . . . . . . . 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 14797 . . . . . . . . . 10  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  S ) )
4746, 2ffvelrnd 5865 . . . . . . . . 9  |-  ( ph  ->  ( ( 1st `  F
) `  X )  e.  ( Base `  S
) )
4847, 41eleqtrrd 2520 . . . . . . . 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 14970 . . . . . 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 2443 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
53 yoneda.1 . . . . . . . 8  |-  .1.  =  ( Id `  C )
5420, 52, 53, 34, 2catidcl 14641 . . . . . . 7  |-  ( ph  ->  (  .1.  `  X
)  e.  ( X ( Hom  `  C
) X ) )
5533, 20, 34, 2, 52, 2yon11 15095 . . . . . . 7  |-  ( ph  ->  ( ( 1st `  (
( 1st `  Y
) `  X )
) `  X )  =  ( X ( Hom  `  C ) X ) )
5654, 55eleqtrrd 2520 . . . . . 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 5865 . . . 4  |-  ( (
ph  /\  a  e.  ( ( ( 1st `  Y ) `  X
) ( O Nat  S
) F ) )  ->  ( ( a `
 X ) `  (  .1.  `  X )
)  e.  ( ( 1st `  F ) `
 X ) )
59 eqid 2443 . . . 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 5888 . . 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 15104 . . . 4  |-  ( ph  ->  ( F ( 1st `  Z ) X )  =  ( ( ( 1st `  Y ) `
 X ) ( O Nat  S ) F ) )
6819oppccat 14682 . . . . . 6  |-  ( C  e.  Cat  ->  O  e.  Cat )
6934, 68syl 16 . . . . 5  |-  ( ph  ->  O  e.  Cat )
7025setccat 14974 . . . . . 6  |-  ( U  e.  _V  ->  S  e.  Cat )
7129, 70syl 16 . . . . 5  |-  ( ph  ->  S  e.  Cat )
7265, 69, 71, 21, 1, 2evlf1 15051 . . . 4  |-  ( ph  ->  ( F ( 1st `  E ) X )  =  ( ( 1st `  F ) `  X
) )
7315, 67, 72feq123d 5570 . . 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 1369    e. wcel 1756   _Vcvv 2993    u. cun 3347    C_ wss 3349   <.cop 3904   class class class wbr 4313    e. cmpt 4371   ran crn 4862   Rel wrel 4866   -->wf 5435   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   1stc1st 6596   2ndc2nd 6597  tpos ctpos 6765   Basecbs 14195   Hom chom 14270   Catccat 14623   Idccid 14624   Hom f chomf 14625  oppCatcoppc 14671    Func cfunc 14785    o.func ccofu 14787   Nat cnat 14872   FuncCat cfuc 14873   SetCatcsetc 14964    X.c cxpc 14999    1stF c1stf 15000    2ndF c2ndf 15001   ⟨,⟩F cprf 15002   evalF cevlf 15040  HomFchof 15079  Yoncyon 15080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-ixp 7285  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-hom 14283  df-cco 14284  df-cat 14627  df-cid 14628  df-homf 14629  df-comf 14630  df-oppc 14672  df-func 14789  df-cofu 14791  df-nat 14874  df-fuc 14875  df-setc 14965  df-xpc 15003  df-1stf 15004  df-2ndf 15005  df-prf 15006  df-evlf 15044  df-curf 15045  df-hof 15081  df-yon 15082
This theorem is referenced by:  yonedalem3b  15110  yonedalem3  15111  yonedainv  15112
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