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Theorem idfu1st 13753
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 2283 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
51, 2, 3, 4idfuval 13750 . . 3  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. )
65fveq2d 5529 . 2  |-  ( ph  ->  ( 1st `  I
)  =  ( 1st `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. ) )
7 fvex 5539 . . . . 5  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2353 . . . 4  |-  B  e. 
_V
9 resiexg 4997 . . . 4  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
108, 9ax-mp 8 . . 3  |-  (  _I  |`  B )  e.  _V
118, 8xpex 4801 . . . 4  |-  ( B  X.  B )  e. 
_V
1211mptex 5746 . . 3  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( (  Hom  `  C
) `  z )
) )  e.  _V
1310, 12op1st 6128 . 2  |-  ( 1st `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. )  =  (  _I  |`  B )
146, 13syl6eq 2331 1  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255   1stc1st 6120   Basecbs 13148    Hom chom 13219   Catccat 13566  idfunccidfu 13729
This theorem is referenced by:  idfu1  13754  cofulid  13764  cofurid  13765  catciso  13939  curf2ndf  14021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-idfu 13733
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