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Theorem df1stres 27345
Description: Definition for a restriction of the  1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
Assertion
Ref Expression
df1stres  |-  ( 1st  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  x )
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem df1stres
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df1st2 6881 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
21reseq1i 5275 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x }  |`  ( A  X.  B
) )  =  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )
3 resoprab 6393 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x }  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  x ) }
4 resres 5292 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 1st  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B
) ) )
5 incom 3696 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  ( A  X.  B ) )
6 xpss 5115 . . . . . . 7  |-  ( A  X.  B )  C_  ( _V  X.  _V )
7 df-ss 3495 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( _V  X.  _V )  <->  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( A  X.  B ) )
86, 7mpbi 208 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( A  X.  B )
95, 8eqtr3i 2498 . . . . 5  |-  ( ( _V  X.  _V )  i^i  ( A  X.  B
) )  =  ( A  X.  B )
109reseq2i 5276 . . . 4  |-  ( 1st  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 1st  |`  ( A  X.  B
) )
114, 10eqtri 2496 . . 3  |-  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 1st  |`  ( A  X.  B ) )
122, 3, 113eqtr3ri 2505 . 2  |-  ( 1st  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  x ) }
13 df-mpt2 6300 . 2  |-  ( x  e.  A ,  y  e.  B  |->  x )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  x
) }
1412, 13eqtr4i 2499 1  |-  ( 1st  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481    X. cxp 5003    |` cres 5007   {coprab 6296    |-> cmpt2 6297   1stc1st 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796
This theorem is referenced by:  cnre2csqima  27718
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