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Theorem idfuval 14065
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 5734 . . . . . 6  |-  ( Base `  c )  e.  _V
43a1i 11 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  e. 
_V )
5 fveq2 5720 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
6 idfuval.b . . . . . 6  |-  B  =  ( Base `  C
)
75, 6syl6eqr 2485 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
8 simpr 448 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  b  =  B )
98reseq2d 5138 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  b
)  =  (  _I  |`  B ) )
108, 8xpeq12d 4895 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  ( b  X.  b
)  =  ( B  X.  B ) )
11 simpl 444 . . . . . . . . . . 11  |-  ( ( c  =  C  /\  b  =  B )  ->  c  =  C )
1211fveq2d 5724 . . . . . . . . . 10  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  (  Hom  `  C ) )
13 idfuval.h . . . . . . . . . 10  |-  H  =  (  Hom  `  C
)
1412, 13syl6eqr 2485 . . . . . . . . 9  |-  ( ( c  =  C  /\  b  =  B )  ->  (  Hom  `  c
)  =  H )
1514fveq1d 5722 . . . . . . . 8  |-  ( ( c  =  C  /\  b  =  B )  ->  ( (  Hom  `  c
) `  z )  =  ( H `  z ) )
1615reseq2d 5138 . . . . . . 7  |-  ( ( c  =  C  /\  b  =  B )  ->  (  _I  |`  (
(  Hom  `  c ) `
 z ) )  =  (  _I  |`  ( H `  z )
) )
1710, 16mpteq12dv 4279 . . . . . 6  |-  ( ( c  =  C  /\  b  =  B )  ->  ( z  e.  ( b  X.  b ) 
|->  (  _I  |`  (
(  Hom  `  c ) `
 z ) ) )  =  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( H `  z
) ) ) )
189, 17opeq12d 3984 . . . . 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 3286 . . . 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 14048 . . . 4  |- idfunc  =  ( c  e. 
Cat  |->  [_ ( Base `  c
)  /  b ]_ <. (  _I  |`  b
) ,  ( z  e.  ( b  X.  b )  |->  (  _I  |`  ( (  Hom  `  c
) `  z )
) ) >. )
21 opex 4419 . . . 4  |-  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  ( H `  z )
) ) >.  e.  _V
2219, 20, 21fvmpt 5798 . . 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 2479 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 1652    e. wcel 1725   _Vcvv 2948   [_csb 3243   <.cop 3809    e. cmpt 4258    _I cid 4485    X. cxp 4868    |` cres 4872   ` cfv 5446   Basecbs 13461    Hom chom 13532   Catccat 13881  idfunccidfu 14044
This theorem is referenced by:  idfu2nd  14066  idfu1st  14068  idfucl  14070  catcisolem  14253  curf2ndf  14336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-idfu 14048
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