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Theorem ida2 14204
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 14203 . . 3  |-  ( ph  ->  ( I `  X
)  =  <. X ,  X ,  (  .1.  `  X ) >. )
76fveq2d 5724 . 2  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  ( 2nd `  <. X ,  X ,  (  .1.  `  X
) >. ) )
8 fvex 5734 . . 3  |-  (  .1.  `  X )  e.  _V
9 ot3rdg 6355 . . 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 2483 1  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  (  .1.  `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cotp 3810   ` cfv 5446   2ndc2nd 6340   Basecbs 13459   Catccat 13879   Idccid 13880  Idacida 14198
This theorem is referenced by:  arwlid  14217  arwrid  14218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-ot 3816  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-2nd 6342  df-ida 14200
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