MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  idaval Structured version   Unicode version

Theorem idaval 15037
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 15036 . 2  |-  ( ph  ->  I  =  ( x  e.  B  |->  <. x ,  x ,  (  .1.  `  x ) >. )
)
6 simpr 461 . . 3  |-  ( (
ph  /\  x  =  X )  ->  x  =  X )
76fveq2d 5796 . . 3  |-  ( (
ph  /\  x  =  X )  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
86, 6, 7oteq123d 4175 . 2  |-  ( (
ph  /\  x  =  X )  ->  <. x ,  x ,  (  .1.  `  x ) >.  =  <. X ,  X ,  (  .1.  `  X ) >. )
9 idaval.x . 2  |-  ( ph  ->  X  e.  B )
10 otex 4658 . . 3  |-  <. X ,  X ,  (  .1.  `  X ) >.  e.  _V
1110a1i 11 . 2  |-  ( ph  -> 
<. X ,  X , 
(  .1.  `  X
) >.  e.  _V )
125, 8, 9, 11fvmptd 5881 1  |-  ( ph  ->  ( I `  X
)  =  <. X ,  X ,  (  .1.  `  X ) >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3071   <.cotp 3986   ` cfv 5519   Basecbs 14285   Catccat 14713   Idccid 14714  Idacida 15032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-ot 3987  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ida 15034
This theorem is referenced by:  ida2  15038  idahom  15039
  Copyright terms: Public domain W3C validator