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Theorem idfu1 13754
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 13753 . . 3  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  B ) )
54fveq1d 5527 . 2  |-  ( ph  ->  ( ( 1st `  I
) `  X )  =  ( (  _I  |`  B ) `  X
) )
6 idfu1.x . . 3  |-  ( ph  ->  X  e.  B )
7 fvresi 5711 . . 3  |-  ( X  e.  B  ->  (
(  _I  |`  B ) `
 X )  =  X )
86, 7syl 15 . 2  |-  ( ph  ->  ( (  _I  |`  B ) `
 X )  =  X )
95, 8eqtrd 2315 1  |-  ( ph  ->  ( ( 1st `  I
) `  X )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    _I cid 4304    |` cres 4691   ` cfv 5255   1stc1st 6120   Basecbs 13148   Catccat 13566  idfunccidfu 13729
This theorem is referenced by:  idffth  13807  ressffth  13812  catciso  13939
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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