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Theorem idafval 13905
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 5541 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 idafval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2346 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5541 . . . . . . . 8  |-  ( c  =  C  ->  ( Id `  c )  =  ( Id `  C
) )
7 idafval.1 . . . . . . . 8  |-  .1.  =  ( Id `  C )
86, 7syl6eqr 2346 . . . . . . 7  |-  ( c  =  C  ->  ( Id `  c )  =  .1.  )
98fveq1d 5543 . . . . . 6  |-  ( c  =  C  ->  (
( Id `  c
) `  x )  =  (  .1.  `  x
) )
109oteq3d 3826 . . . . 5  |-  ( c  =  C  ->  <. x ,  x ,  ( ( Id `  c ) `
 x ) >.  =  <. x ,  x ,  (  .1.  `  x
) >. )
115, 10mpteq12dv 4114 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( Base `  c )  |->  <. x ,  x ,  ( ( Id `  c ) `
 x ) >.
)  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
12 df-ida 13903 . . . 4  |- Ida  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c
)  |->  <. x ,  x ,  ( ( Id
`  c ) `  x ) >. )
)
13 fvex 5555 . . . . . 6  |-  ( Base `  C )  e.  _V
144, 13eqeltri 2366 . . . . 5  |-  B  e. 
_V
1514mptex 5762 . . . 4  |-  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )  e.  _V
1611, 12, 15fvmpt 5618 . . 3  |-  ( C  e.  Cat  ->  (Ida `  C
)  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
172, 16syl 15 . 2  |-  ( ph  ->  (Ida
`  C )  =  ( x  e.  B  |-> 
<. x ,  x ,  (  .1.  `  x
) >. ) )
181, 17syl5eq 2340 1  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cotp 3657    e. cmpt 4093   ` cfv 5271   Basecbs 13164   Catccat 13582   Idccid 13583  Idacida 13901
This theorem is referenced by:  idaval  13906  idaf  13911
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-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-pr 4230
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-sn 3659  df-pr 3660  df-op 3662  df-ot 3663  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-ida 13903
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