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Theorem idfuval 14798
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 5713 . . . . . 6  |-  ( Base `  c )  e.  _V
43a1i 11 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  e. 
_V )
5 fveq2 5703 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
6 idfuval.b . . . . . 6  |-  B  =  ( Base `  C
)
75, 6syl6eqr 2493 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
8 simpr 461 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  b  =  B )
98reseq2d 5122 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  b
)  =  (  _I  |`  B ) )
108, 8xpeq12d 4877 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  ( b  X.  b
)  =  ( B  X.  B ) )
11 simpl 457 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  b  =  B )  ->  c  =  C )
1211fveq2d 5707 . . . . . . . . . 10  |-  ( ( c  =  C  /\  b  =  B )  ->  ( Hom  `  c
)  =  ( Hom  `  C ) )
13 idfuval.h . . . . . . . . . 10  |-  H  =  ( Hom  `  C
)
1412, 13syl6eqr 2493 . . . . . . . . 9  |-  ( ( c  =  C  /\  b  =  B )  ->  ( Hom  `  c
)  =  H )
1514fveq1d 5705 . . . . . . . 8  |-  ( ( c  =  C  /\  b  =  B )  ->  ( ( Hom  `  c
) `  z )  =  ( H `  z ) )
1615reseq2d 5122 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  (
( Hom  `  c ) `
 z ) )  =  (  _I  |`  ( H `  z )
) )
1710, 16mpteq12dv 4382 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  ( z  e.  ( b  X.  b ) 
|->  (  _I  |`  (
( Hom  `  c ) `
 z ) ) )  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
189, 17opeq12d 4079 . . . . 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 3327 . . . 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 14781 . . . 4  |- idfunc  =  ( c  e. 
Cat  |->  [_ ( Base `  c
)  /  b ]_ <. (  _I  |`  b
) ,  ( z  e.  ( b  X.  b )  |->  (  _I  |`  ( ( Hom  `  c
) `  z )
) ) >. )
21 opex 4568 . . . 4  |-  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >.  e.  _V
2219, 20, 21fvmpt 5786 . . 3  |-  ( C  e.  Cat  ->  (idfunc `  C
)  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
232, 22syl 16 . 2  |-  ( ph  ->  (idfunc `  C )  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  ( H `  z )
) ) >. )
241, 23syl5eq 2487 1  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2984   [_csb 3300   <.cop 3895    e. cmpt 4362    _I cid 4643    X. cxp 4850    |` cres 4854   ` cfv 5430   Basecbs 14186   Hom chom 14261   Catccat 14614  idfunccidfu 14777
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  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 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-res 4864  df-iota 5393  df-fun 5432  df-fv 5438  df-idfu 14781
This theorem is referenced by:  idfu2nd  14799  idfu1st  14801  idfucl  14803  catcisolem  14986  curf2ndf  15069
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