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Theorem ida2 15552
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 15551 . . 3  |-  ( ph  ->  ( I `  X
)  =  <. X ,  X ,  (  .1.  `  X ) >. )
76fveq2d 5807 . 2  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  ( 2nd `  <. X ,  X ,  (  .1.  `  X
) >. ) )
8 fvex 5813 . . 3  |-  (  .1.  `  X )  e.  _V
9 ot3rdg 6752 . . 3  |-  ( (  .1.  `  X )  e.  _V  ->  ( 2nd ` 
<. X ,  X , 
(  .1.  `  X
) >. )  =  (  .1.  `  X )
)
108, 9ax-mp 5 . 2  |-  ( 2nd `  <. X ,  X ,  (  .1.  `  X
) >. )  =  (  .1.  `  X )
117, 10syl6eq 2457 1  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  (  .1.  `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    e. wcel 1840   _Vcvv 3056   <.cotp 3977   ` cfv 5523   2ndc2nd 6735   Basecbs 14731   Catccat 15168   Idccid 15169  Idacida 15546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-ot 3978  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-2nd 6737  df-ida 15548
This theorem is referenced by:  arwlid  15565  arwrid  15566
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