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Theorem idfu1st 15308
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 2396 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
51, 2, 3, 4idfuval 15305 . . 3  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  (
( Hom  `  C ) `
 z ) ) ) >. )
65fveq2d 5795 . 2  |-  ( ph  ->  ( 1st `  I
)  =  ( 1st `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  (
( Hom  `  C ) `
 z ) ) ) >. ) )
7 fvex 5801 . . . . 5  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2480 . . . 4  |-  B  e. 
_V
9 resiexg 6657 . . . 4  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
108, 9ax-mp 5 . . 3  |-  (  _I  |`  B )  e.  _V
118, 8xpex 6525 . . . 4  |-  ( B  X.  B )  e. 
_V
1211mptex 6064 . . 3  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( ( Hom  `  C
) `  z )
) )  e.  _V
1310, 12op1st 6729 . 2  |-  ( 1st `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  (
( Hom  `  C ) `
 z ) ) ) >. )  =  (  _I  |`  B )
146, 13syl6eq 2453 1  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1399    e. wcel 1836   _Vcvv 3051   <.cop 3967    |-> cmpt 4442    _I cid 4721    X. cxp 4928    |` cres 4932   ` cfv 5513   1stc1st 6719   Basecbs 14657   Hom chom 14736   Catccat 15094  idfunccidfu 15284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-rep 4495  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-1st 6721  df-idfu 15288
This theorem is referenced by:  idfu1  15309  cofulid  15319  cofurid  15320  catciso  15526  curf2ndf  15656
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