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Theorem df1stres 23934
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 6374 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
21reseq1i 5084 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x }  |`  ( A  X.  B
) )  =  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )
3 resoprab 6107 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x }  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  x ) }
4 resres 5101 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 1st  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B
) ) )
5 incom 3478 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  ( A  X.  B ) )
6 xpss 4924 . . . . . . 7  |-  ( A  X.  B )  C_  ( _V  X.  _V )
7 df-ss 3279 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( _V  X.  _V )  <->  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( A  X.  B ) )
86, 7mpbi 200 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( A  X.  B )
95, 8eqtr3i 2411 . . . . 5  |-  ( ( _V  X.  _V )  i^i  ( A  X.  B
) )  =  ( A  X.  B )
109reseq2i 5085 . . . 4  |-  ( 1st  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 1st  |`  ( A  X.  B
) )
114, 10eqtri 2409 . . 3  |-  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 1st  |`  ( A  X.  B ) )
122, 3, 113eqtr3ri 2418 . 2  |-  ( 1st  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  x ) }
13 df-mpt2 6027 . 2  |-  ( x  e.  A ,  y  e.  B  |->  x )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  x
) }
1412, 13eqtr4i 2412 1  |-  ( 1st  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  x )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2901    i^i cin 3264    C_ wss 3265    X. cxp 4818    |` cres 4822   {coprab 6023    e. cmpt2 6024   1stc1st 6288
This theorem is referenced by:  cnre2csqima  24115
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291
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