MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  idfuval Unicode version

Theorem idfuval 13750
Description: Value of the composition of two functors. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
idfuval.i  |-  I  =  (idfunc `  C )
idfuval.b  |-  B  =  ( Base `  C
)
idfuval.c  |-  ( ph  ->  C  e.  Cat )
idfuval.h  |-  H  =  (  Hom  `  C
)
Assertion
Ref Expression
idfuval  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
Distinct variable groups:    z, B    z, C    z, H    ph, z
Allowed substitution hint:    I( z)

Proof of Theorem idfuval
Dummy variables  b 
c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idfuval.i . 2  |-  I  =  (idfunc `  C )
2 idfuval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 fvex 5539 . . . . . 6  |-  ( Base `  c )  e.  _V
43a1i 10 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  e. 
_V )
5 fveq2 5525 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
6 idfuval.b . . . . . 6  |-  B  =  ( Base `  C
)
75, 6syl6eqr 2333 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
8 simpr 447 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  b  =  B )
98reseq2d 4955 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  b
)  =  (  _I  |`  B ) )
108, 8xpeq12d 4714 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  ( b  X.  b
)  =  ( B  X.  B ) )
11 simpl 443 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  b  =  B )  ->  c  =  C )
1211fveq2d 5529 . . . . . . . . . 10  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  (  Hom  `  C ) )
13 idfuval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
1412, 13syl6eqr 2333 . . . . . . . . 9  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  H )
1514fveq1d 5527 . . . . . . . 8  |-  ( ( c  =  C  /\  b  =  B )  ->  ( (  Hom  `  c
) `  z )  =  ( H `  z ) )
1615reseq2d 4955 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  (
(  Hom  `  c ) `
 z ) )  =  (  _I  |`  ( H `  z )
) )
1710, 16mpteq12dv 4098 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  ( z  e.  ( b  X.  b ) 
|->  (  _I  |`  (
(  Hom  `  c ) `
 z ) ) )  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
189, 17opeq12d 3804 . . . . 5  |-  ( ( c  =  C  /\  b  =  B )  -> 
<. (  _I  |`  b
) ,  ( z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  c
) `  z )
) ) >.  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
194, 7, 18csbied2 3124 . . . 4  |-  ( c  =  C  ->  [_ ( Base `  c )  / 
b ]_ <. (  _I  |`  b
) ,  ( z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  c
) `  z )
) ) >.  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
20 df-idfu 13733 . . . 4  |- idfunc  =  ( c  e. 
Cat  |->  [_ ( Base `  c
)  /  b ]_ <. (  _I  |`  b
) ,  ( z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  c
) `  z )
) ) >. )
21 opex 4237 . . . 4  |-  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >.  e.  _V
2219, 20, 21fvmpt 5602 . . 3  |-  ( C  e.  Cat  ->  (idfunc `  C
)  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
232, 22syl 15 . 2  |-  ( ph  ->  (idfunc `  C )  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
241, 23syl5eq 2327 1  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   [_csb 3081   <.cop 3643    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255   Basecbs 13148    Hom chom 13219   Catccat 13566  idfunccidfu 13729
This theorem is referenced by:  idfu2nd  13751  idfu1st  13753  idfucl  13755  catcisolem  13938  curf2ndf  14021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-idfu 13733
  Copyright terms: Public domain W3C validator