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Theorem df2ndres 28278
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 6880 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
21reseq1i 5100 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y }  |`  ( A  X.  B
) )  =  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )
3 resoprab 6389 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y }  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  y ) }
4 resres 5116 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 2nd  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B
) ) )
5 incom 3624 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  ( A  X.  B ) )
6 xpss 4940 . . . . . . 7  |-  ( A  X.  B )  C_  ( _V  X.  _V )
7 df-ss 3417 . . . . . . 7  |-  ( ( A  X.  B ) 
C_  ( _V  X.  _V )  <->  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( A  X.  B ) )
86, 7mpbi 212 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( A  X.  B )
95, 8eqtr3i 2474 . . . . 5  |-  ( ( _V  X.  _V )  i^i  ( A  X.  B
) )  =  ( A  X.  B )
109reseq2i 5101 . . . 4  |-  ( 2nd  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 2nd  |`  ( A  X.  B
) )
114, 10eqtri 2472 . . 3  |-  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 2nd  |`  ( A  X.  B ) )
122, 3, 113eqtr3ri 2481 . 2  |-  ( 2nd  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  y ) }
13 df-mpt2 6293 . 2  |-  ( x  e.  A ,  y  e.  B  |->  y )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  y
) }
1412, 13eqtr4i 2475 1  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  y )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1443    e. wcel 1886   _Vcvv 3044    i^i cin 3402    C_ wss 3403    X. cxp 4831    |` cres 4835   {coprab 6289    |-> cmpt2 6290   2ndc2nd 6789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791
This theorem is referenced by:  cnre2csqima  28710
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