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Theorem idafval 14140
Description: Value of the identity arrow function. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
idafval.i  |-  I  =  (Ida
`  C )
idafval.b  |-  B  =  ( Base `  C
)
idafval.c  |-  ( ph  ->  C  e.  Cat )
idafval.1  |-  .1.  =  ( Id `  C )
Assertion
Ref Expression
idafval  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
Distinct variable groups:    x,  .1.    x, B    x, C    x, I    ph, x

Proof of Theorem idafval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 idafval.i . 2  |-  I  =  (Ida
`  C )
2 idafval.c . . 3  |-  ( ph  ->  C  e.  Cat )
3 fveq2 5669 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 idafval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2438 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5669 . . . . . . . 8  |-  ( c  =  C  ->  ( Id `  c )  =  ( Id `  C
) )
7 idafval.1 . . . . . . . 8  |-  .1.  =  ( Id `  C )
86, 7syl6eqr 2438 . . . . . . 7  |-  ( c  =  C  ->  ( Id `  c )  =  .1.  )
98fveq1d 5671 . . . . . 6  |-  ( c  =  C  ->  (
( Id `  c
) `  x )  =  (  .1.  `  x
) )
109oteq3d 3941 . . . . 5  |-  ( c  =  C  ->  <. x ,  x ,  ( ( Id `  c ) `
 x ) >.  =  <. x ,  x ,  (  .1.  `  x
) >. )
115, 10mpteq12dv 4229 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( Base `  c )  |->  <. x ,  x ,  ( ( Id `  c ) `
 x ) >.
)  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
12 df-ida 14138 . . . 4  |- Ida  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c
)  |->  <. x ,  x ,  ( ( Id
`  c ) `  x ) >. )
)
13 fvex 5683 . . . . . 6  |-  ( Base `  C )  e.  _V
144, 13eqeltri 2458 . . . . 5  |-  B  e. 
_V
1514mptex 5906 . . . 4  |-  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )  e.  _V
1611, 12, 15fvmpt 5746 . . 3  |-  ( C  e.  Cat  ->  (Ida `  C
)  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
172, 16syl 16 . 2  |-  ( ph  ->  (Ida
`  C )  =  ( x  e.  B  |-> 
<. x ,  x ,  (  .1.  `  x
) >. ) )
181, 17syl5eq 2432 1  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2900   <.cotp 3762    e. cmpt 4208   ` cfv 5395   Basecbs 13397   Catccat 13817   Idccid 13818  Idacida 14136
This theorem is referenced by:  idaval  14141  idaf  14146
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pr 4345
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-ot 3768  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ida 14138
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