MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ida2 Unicode version

Theorem ida2 14143
Description: Morphism part of the identity arrow. (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
ida2  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  (  .1.  `  X ) )

Proof of Theorem ida2
StepHypRef Expression
1 idafval.i . . . 4  |-  I  =  (Ida
`  C )
2 idafval.b . . . 4  |-  B  =  ( Base `  C
)
3 idafval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 idafval.1 . . . 4  |-  .1.  =  ( Id `  C )
5 idaval.x . . . 4  |-  ( ph  ->  X  e.  B )
61, 2, 3, 4, 5idaval 14142 . . 3  |-  ( ph  ->  ( I `  X
)  =  <. X ,  X ,  (  .1.  `  X ) >. )
76fveq2d 5674 . 2  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  ( 2nd `  <. X ,  X ,  (  .1.  `  X
) >. ) )
8 fvex 5684 . . 3  |-  (  .1.  `  X )  e.  _V
9 ot3rdg 6304 . . 3  |-  ( (  .1.  `  X )  e.  _V  ->  ( 2nd ` 
<. X ,  X , 
(  .1.  `  X
) >. )  =  (  .1.  `  X )
)
108, 9ax-mp 8 . 2  |-  ( 2nd `  <. X ,  X ,  (  .1.  `  X
) >. )  =  (  .1.  `  X )
117, 10syl6eq 2437 1  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  (  .1.  `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2901   <.cotp 3763   ` cfv 5396   2ndc2nd 6289   Basecbs 13398   Catccat 13818   Idccid 13819  Idacida 14137
This theorem is referenced by:  arwlid  14156  arwrid  14157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-ot 3769  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-2nd 6291  df-ida 14139
  Copyright terms: Public domain W3C validator