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Theorem idaval 15536
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 15535 . 2  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
6 simpr 459 . . 3  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
76fveq2d 5852 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
86, 6, 7oteq123d 4218 . 2  |-  ( (
ph  /\  x  =  X )  ->  <. x ,  x ,  (  .1.  `  x ) >.  =  <. X ,  X ,  (  .1.  `  X ) >. )
9 idaval.x . 2  |-  ( ph  ->  X  e.  B )
10 otex 4702 . . 3  |-  <. X ,  X ,  (  .1.  `  X ) >.  e.  _V
1110a1i 11 . 2  |-  ( ph  -> 
<. X ,  X , 
(  .1.  `  X
) >.  e.  _V )
125, 8, 9, 11fvmptd 5936 1  |-  ( ph  ->  ( I `  X
)  =  <. X ,  X ,  (  .1.  `  X ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   <.cotp 4024   ` cfv 5570   Basecbs 14716   Catccat 15153   Idccid 15154  Idacida 15531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-ot 4025  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ida 15533
This theorem is referenced by:  ida2  15537  idahom  15538
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