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Theorem idfu1st 14076
Description: Value of the object part of the identity functor. (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 )
Assertion
Ref Expression
idfu1st  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  B ) )

Proof of Theorem idfu1st
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 idfuval.i . . . 4  |-  I  =  (idfunc `  C )
2 idfuval.b . . . 4  |-  B  =  ( Base `  C
)
3 idfuval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 eqid 2436 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
51, 2, 3, 4idfuval 14073 . . 3  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. )
65fveq2d 5732 . 2  |-  ( ph  ->  ( 1st `  I
)  =  ( 1st `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. ) )
7 fvex 5742 . . . . 5  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2506 . . . 4  |-  B  e. 
_V
9 resiexg 5188 . . . 4  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
108, 9ax-mp 8 . . 3  |-  (  _I  |`  B )  e.  _V
118, 8xpex 4990 . . . 4  |-  ( B  X.  B )  e. 
_V
1211mptex 5966 . . 3  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( (  Hom  `  C
) `  z )
) )  e.  _V
1310, 12op1st 6355 . 2  |-  ( 1st `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. )  =  (  _I  |`  B )
146, 13syl6eq 2484 1  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956   <.cop 3817    e. cmpt 4266    _I cid 4493    X. cxp 4876    |` cres 4880   ` cfv 5454   1stc1st 6347   Basecbs 13469    Hom chom 13540   Catccat 13889  idfunccidfu 14052
This theorem is referenced by:  idfu1  14077  cofulid  14087  cofurid  14088  catciso  14262  curf2ndf  14344
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1st 6349  df-idfu 14056
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