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Theorem idfu1 14004
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 )
idfu1.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
idfu1  |-  ( ph  ->  ( ( 1st `  I
) `  X )  =  X )

Proof of Theorem idfu1
StepHypRef Expression
1 idfuval.i . . . 4  |-  I  =  (idfunc `  C )
2 idfuval.b . . . 4  |-  B  =  ( Base `  C
)
3 idfuval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
41, 2, 3idfu1st 14003 . . 3  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  B ) )
54fveq1d 5670 . 2  |-  ( ph  ->  ( ( 1st `  I
) `  X )  =  ( (  _I  |`  B ) `  X
) )
6 idfu1.x . . 3  |-  ( ph  ->  X  e.  B )
7 fvresi 5863 . . 3  |-  ( X  e.  B  ->  (
(  _I  |`  B ) `
 X )  =  X )
86, 7syl 16 . 2  |-  ( ph  ->  ( (  _I  |`  B ) `
 X )  =  X )
95, 8eqtrd 2419 1  |-  ( ph  ->  ( ( 1st `  I
) `  X )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    _I cid 4434    |` cres 4820   ` cfv 5394   1stc1st 6286   Basecbs 13396   Catccat 13816  idfunccidfu 13979
This theorem is referenced by:  idffth  14057  ressffth  14062  catciso  14189
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-1st 6288  df-idfu 13983
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