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Theorem idaval 13890
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 )
idaval.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
idaval  |-  ( ph  ->  ( I `  X
)  =  <. X ,  X ,  (  .1.  `  X ) >. )

Proof of Theorem idaval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 idafval.i . . 3  |-  I  =  (Ida
`  C )
2 idafval.b . . 3  |-  B  =  ( Base `  C
)
3 idafval.c . . 3  |-  ( ph  ->  C  e.  Cat )
4 idafval.1 . . 3  |-  .1.  =  ( Id `  C )
51, 2, 3, 4idafval 13889 . 2  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
6 simpr 447 . . 3  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
76fveq2d 5529 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
86, 6, 7oteq123d 3811 . 2  |-  ( (
ph  /\  x  =  X )  ->  <. x ,  x ,  (  .1.  `  x ) >.  =  <. X ,  X ,  (  .1.  `  X ) >. )
9 idaval.x . 2  |-  ( ph  ->  X  e.  B )
10 otex 4238 . . 3  |-  <. X ,  X ,  (  .1.  `  X ) >.  e.  _V
1110a1i 10 . 2  |-  ( ph  -> 
<. X ,  X , 
(  .1.  `  X
) >.  e.  _V )
125, 8, 9, 11fvmptd 5606 1  |-  ( ph  ->  ( I `  X
)  =  <. X ,  X ,  (  .1.  `  X ) >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cotp 3644   ` cfv 5255   Basecbs 13148   Catccat 13566   Idccid 13567  Idacida 13885
This theorem is referenced by:  ida2  13891  idahom  13892
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-ot 3650  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ida 13887
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