Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df1stres Structured version   Unicode version

Theorem df1stres 27669
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 6785 . . . 4  |-  { <. <.
x ,  y >. ,  z >.  |  z  =  x }  =  ( 1st  |`  ( _V  X.  _V ) )
21reseq1i 5182 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x }  |`  ( A  X.  B
) )  =  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )
3 resoprab 6297 . . 3  |-  ( {
<. <. x ,  y
>. ,  z >.  |  z  =  x }  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  x ) }
4 resres 5198 . . . 4  |-  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 1st  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B
) ) )
5 incom 3605 . . . . . 6  |-  ( ( A  X.  B )  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  ( A  X.  B ) )
6 xpss 5022 . . . . . . 7  |-  ( A  X.  B )  C_  ( _V  X.  _V )
7 df-ss 3403 . . . . . . 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 2413 . . . . 5  |-  ( ( _V  X.  _V )  i^i  ( A  X.  B
) )  =  ( A  X.  B )
109reseq2i 5183 . . . 4  |-  ( 1st  |`  ( ( _V  X.  _V )  i^i  ( A  X.  B ) ) )  =  ( 1st  |`  ( A  X.  B
) )
114, 10eqtri 2411 . . 3  |-  ( ( 1st  |`  ( _V  X.  _V ) )  |`  ( A  X.  B
) )  =  ( 1st  |`  ( A  X.  B ) )
122, 3, 113eqtr3ri 2420 . 2  |-  ( 1st  |`  ( A  X.  B
) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  z  =  x ) }
13 df-mpt2 6201 . 2  |-  ( x  e.  A ,  y  e.  B  |->  x )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  x
) }
1412, 13eqtr4i 2414 1  |-  ( 1st  |`  ( A  X.  B
) )  =  ( x  e.  A , 
y  e.  B  |->  x )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    = wceq 1399    e. wcel 1826   _Vcvv 3034    i^i cin 3388    C_ wss 3389    X. cxp 4911    |` cres 4915   {coprab 6197    |-> cmpt2 6198   1stc1st 6697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700
This theorem is referenced by:  cnre2csqima  28047
  Copyright terms: Public domain W3C validator