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Theorem idfuval 13766
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 5555 . . . . . 6  |-  ( Base `  c )  e.  _V
43a1i 10 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  e. 
_V )
5 fveq2 5541 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
6 idfuval.b . . . . . 6  |-  B  =  ( Base `  C
)
75, 6syl6eqr 2346 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
8 simpr 447 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  b  =  B )
98reseq2d 4971 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  b
)  =  (  _I  |`  B ) )
108, 8xpeq12d 4730 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  ( b  X.  b
)  =  ( B  X.  B ) )
11 simpl 443 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  b  =  B )  ->  c  =  C )
1211fveq2d 5545 . . . . . . . . . 10  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  (  Hom  `  C ) )
13 idfuval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
1412, 13syl6eqr 2346 . . . . . . . . 9  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  H )
1514fveq1d 5543 . . . . . . . 8  |-  ( ( c  =  C  /\  b  =  B )  ->  ( (  Hom  `  c
) `  z )  =  ( H `  z ) )
1615reseq2d 4971 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  (
(  Hom  `  c ) `
 z ) )  =  (  _I  |`  ( H `  z )
) )
1710, 16mpteq12dv 4114 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  ( z  e.  ( b  X.  b ) 
|->  (  _I  |`  (
(  Hom  `  c ) `
 z ) ) )  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
189, 17opeq12d 3820 . . . . 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 3137 . . . 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 13749 . . . 4  |- idfunc  =  ( c  e. 
Cat  |->  [_ ( Base `  c
)  /  b ]_ <. (  _I  |`  b
) ,  ( z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  c
) `  z )
) ) >. )
21 opex 4253 . . . 4  |-  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >.  e.  _V
2219, 20, 21fvmpt 5618 . . 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 2340 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 1632    e. wcel 1696   _Vcvv 2801   [_csb 3094   <.cop 3656    e. cmpt 4093    _I cid 4320    X. cxp 4703    |` cres 4707   ` cfv 5271   Basecbs 13164    Hom chom 13235   Catccat 13582  idfunccidfu 13745
This theorem is referenced by:  idfu2nd  13767  idfu1st  13769  idfucl  13771  catcisolem  13954  curf2ndf  14037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-idfu 13749
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