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Theorem idafval 14930
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 5696 . . . . . 6  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
4 idafval.b . . . . . 6  |-  B  =  ( Base `  C
)
53, 4syl6eqr 2493 . . . . 5  |-  ( c  =  C  ->  ( Base `  c )  =  B )
6 fveq2 5696 . . . . . . . 8  |-  ( c  =  C  ->  ( Id `  c )  =  ( Id `  C
) )
7 idafval.1 . . . . . . . 8  |-  .1.  =  ( Id `  C )
86, 7syl6eqr 2493 . . . . . . 7  |-  ( c  =  C  ->  ( Id `  c )  =  .1.  )
98fveq1d 5698 . . . . . 6  |-  ( c  =  C  ->  (
( Id `  c
) `  x )  =  (  .1.  `  x
) )
109oteq3d 4078 . . . . 5  |-  ( c  =  C  ->  <. x ,  x ,  ( ( Id `  c ) `
 x ) >.  =  <. x ,  x ,  (  .1.  `  x
) >. )
115, 10mpteq12dv 4375 . . . 4  |-  ( c  =  C  ->  (
x  e.  ( Base `  c )  |->  <. x ,  x ,  ( ( Id `  c ) `
 x ) >.
)  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
12 df-ida 14928 . . . 4  |- Ida  =  ( c  e.  Cat  |->  ( x  e.  ( Base `  c
)  |->  <. x ,  x ,  ( ( Id
`  c ) `  x ) >. )
)
13 fvex 5706 . . . . . 6  |-  ( Base `  C )  e.  _V
144, 13eqeltri 2513 . . . . 5  |-  B  e. 
_V
1514mptex 5953 . . . 4  |-  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )  e.  _V
1611, 12, 15fvmpt 5779 . . 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 2487 1  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2977   <.cotp 3890    e. cmpt 4355   ` cfv 5423   Basecbs 14179   Catccat 14607   Idccid 14608  Idacida 14926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-ot 3891  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ida 14928
This theorem is referenced by:  idaval  14931  idaf  14936
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