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Theorem idfu1st 13769
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 2296 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
51, 2, 3, 4idfuval 13766 . . 3  |-  ( ph  ->  I  =  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B ) 
|->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. )
65fveq2d 5545 . 2  |-  ( ph  ->  ( 1st `  I
)  =  ( 1st `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. ) )
7 fvex 5555 . . . . 5  |-  ( Base `  C )  e.  _V
82, 7eqeltri 2366 . . . 4  |-  B  e. 
_V
9 resiexg 5013 . . . 4  |-  ( B  e.  _V  ->  (  _I  |`  B )  e. 
_V )
108, 9ax-mp 8 . . 3  |-  (  _I  |`  B )  e.  _V
118, 8xpex 4817 . . . 4  |-  ( B  X.  B )  e. 
_V
1211mptex 5762 . . 3  |-  ( z  e.  ( B  X.  B )  |->  (  _I  |`  ( (  Hom  `  C
) `  z )
) )  e.  _V
1310, 12op1st 6144 . 2  |-  ( 1st `  <. (  _I  |`  B ) ,  ( z  e.  ( B  X.  B
)  |->  (  _I  |`  (
(  Hom  `  C ) `
 z ) ) ) >. )  =  (  _I  |`  B )
146, 13syl6eq 2344 1  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656    e. cmpt 4093    _I cid 4320    X. cxp 4703    |` cres 4707   ` cfv 5271   1stc1st 6136   Basecbs 13164    Hom chom 13235   Catccat 13582  idfunccidfu 13745
This theorem is referenced by:  idfu1  13770  cofulid  13780  cofurid  13781  catciso  13955  curf2ndf  14037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6138  df-idfu 13749
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