Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df2ndres Structured version   Unicode version

Theorem df2ndres 26146
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 6765 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  y }  =  ( 2nd  |`  ( _V  X.  _V ) )
21reseq1i 5209 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y }  |`  ( A  X.  B
) )  =  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )
3 resoprab 6291 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  y }  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  y ) }
4 resres 5226 . . . 4  |-  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 2nd  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B
) ) )
5 incom 3646 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  ( A  X.  B ) )
6 xpss 5049 . . . . . . 7  |-  ( A  X.  B )  C_  ( _V  X.  _V )
7 df-ss 3445 . . . . . . 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 2483 . . . . 5  |-  ( ( _V  X.  _V )  i^i  ( A  X.  B
) )  =  ( A  X.  B )
109reseq2i 5210 . . . 4  |-  ( 2nd  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 2nd  |`  ( A  X.  B
) )
114, 10eqtri 2481 . . 3  |-  ( ( 2nd  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 2nd  |`  ( A  X.  B ) )
122, 3, 113eqtr3ri 2490 . 2  |-  ( 2nd  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  y ) }
13 df-mpt2 6200 . 2  |-  ( x  e.  A ,  y  e.  B  |->  y )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  y
) }
1412, 13eqtr4i 2484 1  |-  ( 2nd  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  y )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3072    i^i cin 3430    C_ wss 3431    X. cxp 4941    |` cres 4945   {coprab 6196    |-> cmpt2 6197   2ndc2nd 6681
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 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
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 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-fo 5527  df-fv 5529  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683
This theorem is referenced by:  cnre2csqima  26481
  Copyright terms: Public domain W3C validator