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Theorem df2ndres 27678
Description: Definition for a restriction of the  2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
Assertion
Ref Expression
df2ndres  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  y )
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem df2ndres
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df2nd2 6886 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
21reseq1i 5279 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y }  |`  ( A  X.  B
) )  =  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )
3 resoprab 6397 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y }  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  y ) }
4 resres 5296 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 2nd  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B
) ) )
5 incom 3687 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  ( A  X.  B ) )
6 xpss 5118 . . . . . . 7  |-  ( A  X.  B )  C_  ( _V  X.  _V )
7 df-ss 3485 . . . . . . 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 2488 . . . . 5  |-  ( ( _V  X.  _V )  i^i  ( A  X.  B
) )  =  ( A  X.  B )
109reseq2i 5280 . . . 4  |-  ( 2nd  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 2nd  |`  ( A  X.  B
) )
114, 10eqtri 2486 . . 3  |-  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 2nd  |`  ( A  X.  B ) )
122, 3, 113eqtr3ri 2495 . 2  |-  ( 2nd  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  y ) }
13 df-mpt2 6301 . 2  |-  ( x  e.  A ,  y  e.  B  |->  y )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  y
) }
1412, 13eqtr4i 2489 1  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  y )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109    i^i cin 3470    C_ wss 3471    X. cxp 5006    |` cres 5010   {coprab 6297    |-> cmpt2 6298   2ndc2nd 6798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800
This theorem is referenced by:  cnre2csqima  28054
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