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Theorem ida2 15049
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 15048 . . 3  |-  ( ph  ->  ( I `  X
)  =  <. X ,  X ,  (  .1.  `  X ) >. )
76fveq2d 5806 . 2  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  ( 2nd `  <. X ,  X ,  (  .1.  `  X
) >. ) )
8 fvex 5812 . . 3  |-  (  .1.  `  X )  e.  _V
9 ot3rdg 6706 . . 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 2511 1  |-  ( ph  ->  ( 2nd `  (
I `  X )
)  =  (  .1.  `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078   <.cotp 3996   ` cfv 5529   2ndc2nd 6689   Basecbs 14295   Catccat 14724   Idccid 14725  Idacida 15043
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-ot 3997  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-2nd 6691  df-ida 15045
This theorem is referenced by:  arwlid  15062  arwrid  15063
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