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Theorem idfu1 14069
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 14068 . . 3  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  B ) )
54fveq1d 5722 . 2  |-  ( ph  ->  ( ( 1st `  I
) `  X )  =  ( (  _I  |`  B ) `  X
) )
6 idfu1.x . . 3  |-  ( ph  ->  X  e.  B )
7 fvresi 5916 . . 3  |-  ( X  e.  B  ->  (
(  _I  |`  B ) `
 X )  =  X )
86, 7syl 16 . 2  |-  ( ph  ->  ( (  _I  |`  B ) `
 X )  =  X )
95, 8eqtrd 2467 1  |-  ( ph  ->  ( ( 1st `  I
) `  X )  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    _I cid 4485    |` cres 4872   ` cfv 5446   1stc1st 6339   Basecbs 13461   Catccat 13881  idfunccidfu 14044
This theorem is referenced by:  idffth  14122  ressffth  14127  catciso  14254
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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-reu 2704  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-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  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-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1st 6341  df-idfu 14048
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