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Theorem df1stres 26135
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 6761 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
21reseq1i 5206 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x }  |`  ( A  X.  B
) )  =  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )
3 resoprab 6288 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x }  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  x ) }
4 resres 5223 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 1st  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B
) ) )
5 incom 3643 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  ( A  X.  B ) )
6 xpss 5046 . . . . . . 7  |-  ( A  X.  B )  C_  ( _V  X.  _V )
7 df-ss 3442 . . . . . . 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 2482 . . . . 5  |-  ( ( _V  X.  _V )  i^i  ( A  X.  B
) )  =  ( A  X.  B )
109reseq2i 5207 . . . 4  |-  ( 1st  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 1st  |`  ( A  X.  B
) )
114, 10eqtri 2480 . . 3  |-  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 1st  |`  ( A  X.  B ) )
122, 3, 113eqtr3ri 2489 . 2  |-  ( 1st  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  x ) }
13 df-mpt2 6197 . 2  |-  ( x  e.  A ,  y  e.  B  |->  x )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  x
) }
1412, 13eqtr4i 2483 1  |-  ( 1st  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3070    i^i cin 3427    C_ wss 3428    X. cxp 4938    |` cres 4942   {coprab 6193    |-> cmpt2 6194   1stc1st 6677
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 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-fo 5524  df-fv 5526  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680
This theorem is referenced by:  cnre2csqima  26477
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