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Theorem idfuval 14001
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 5683 . . . . . 6  |-  ( Base `  c )  e.  _V
43a1i 11 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  e. 
_V )
5 fveq2 5669 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
6 idfuval.b . . . . . 6  |-  B  =  ( Base `  C
)
75, 6syl6eqr 2438 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
8 simpr 448 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  b  =  B )
98reseq2d 5087 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  b
)  =  (  _I  |`  B ) )
108, 8xpeq12d 4844 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  ( b  X.  b
)  =  ( B  X.  B ) )
11 simpl 444 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  b  =  B )  ->  c  =  C )
1211fveq2d 5673 . . . . . . . . . 10  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  (  Hom  `  C ) )
13 idfuval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
1412, 13syl6eqr 2438 . . . . . . . . 9  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  H )
1514fveq1d 5671 . . . . . . . 8  |-  ( ( c  =  C  /\  b  =  B )  ->  ( (  Hom  `  c
) `  z )  =  ( H `  z ) )
1615reseq2d 5087 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  (
(  Hom  `  c ) `
 z ) )  =  (  _I  |`  ( H `  z )
) )
1710, 16mpteq12dv 4229 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  ( z  e.  ( b  X.  b ) 
|->  (  _I  |`  (
(  Hom  `  c ) `
 z ) ) )  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
189, 17opeq12d 3935 . . . . 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 3238 . . . 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 13984 . . . 4  |- idfunc  =  ( c  e. 
Cat  |->  [_ ( Base `  c
)  /  b ]_ <. (  _I  |`  b
) ,  ( z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  c
) `  z )
) ) >. )
21 opex 4369 . . . 4  |-  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >.  e.  _V
2219, 20, 21fvmpt 5746 . . 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 2432 1  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2900   [_csb 3195   <.cop 3761    e. cmpt 4208    _I cid 4435    X. cxp 4817    |` cres 4821   ` cfv 5395   Basecbs 13397    Hom chom 13468   Catccat 13817  idfunccidfu 13980
This theorem is referenced by:  idfu2nd  14002  idfu1st  14004  idfucl  14006  catcisolem  14189  curf2ndf  14272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-res 4831  df-iota 5359  df-fun 5397  df-fv 5403  df-idfu 13984
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