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Theorem xpinpreima 24339
Description: Rewrite the cartesian product of two sets as the intersection of their preimage by  1st and  2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
Assertion
Ref Expression
xpinpreima  |-  ( A  X.  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" B ) )

Proof of Theorem xpinpreima
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 inrab 3601 . 2  |-  ( { r  e.  ( _V 
X.  _V )  |  ( 1st `  r )  e.  A }  i^i  { r  e.  ( _V 
X.  _V )  |  ( 2nd `  r )  e.  B } )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r )  e.  B
) }
2 f1stres 6404 . . . . 5  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
3 ffn 5626 . . . . 5  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
4 fncnvima2 5888 . . . . 5  |-  ( ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V )  ->  ( `' ( 1st  |`  ( _V  X.  _V ) ) " A )  =  {
r  e.  ( _V 
X.  _V )  |  ( ( 1st  |`  ( _V  X.  _V ) ) `
 r )  e.  A } )
52, 3, 4mp2b 10 . . . 4  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" A )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st  |`  ( _V  X.  _V ) ) `  r
)  e.  A }
6 fvres 5776 . . . . . 6  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  r )  =  ( 1st `  r ) )
76eleq1d 2509 . . . . 5  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 r )  e.  A  <->  ( 1st `  r
)  e.  A ) )
87rabbiia 2955 . . . 4  |-  { r  e.  ( _V  X.  _V )  |  (
( 1st  |`  ( _V 
X.  _V ) ) `  r )  e.  A }  =  { r  e.  ( _V  X.  _V )  |  ( 1st `  r )  e.  A }
95, 8eqtri 2463 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" A )  =  { r  e.  ( _V  X.  _V )  |  ( 1st `  r
)  e.  A }
10 f2ndres 6405 . . . . 5  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5626 . . . . 5  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fncnvima2 5888 . . . . 5  |-  ( ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V )  ->  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " B )  =  {
r  e.  ( _V 
X.  _V )  |  ( ( 2nd  |`  ( _V  X.  _V ) ) `
 r )  e.  B } )
1310, 11, 12mp2b 10 . . . 4  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 2nd  |`  ( _V  X.  _V ) ) `  r
)  e.  B }
14 fvres 5776 . . . . . 6  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  r )  =  ( 2nd `  r ) )
1514eleq1d 2509 . . . . 5  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 r )  e.  B  <->  ( 2nd `  r
)  e.  B ) )
1615rabbiia 2955 . . . 4  |-  { r  e.  ( _V  X.  _V )  |  (
( 2nd  |`  ( _V 
X.  _V ) ) `  r )  e.  B }  =  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r )  e.  B }
1713, 16eqtri 2463 . . 3  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" B )  =  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r
)  e.  B }
189, 17ineq12i 3529 . 2  |-  ( ( `' ( 1st  |`  ( _V  X.  _V ) )
" A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " B
) )  =  ( { r  e.  ( _V  X.  _V )  |  ( 1st `  r
)  e.  A }  i^i  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r
)  e.  B }
)
19 xp2 6420 . 2  |-  ( A  X.  B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }
201, 18, 193eqtr4ri 2474 1  |-  ( A  X.  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" B ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 360    = wceq 1654    e. wcel 1728   {crab 2716   _Vcvv 2965    i^i cin 3308    X. cxp 4911   `'ccnv 4912    |` cres 4915   "cima 4916    Fn wfn 5484   -->wf 5485   ` cfv 5489   1stc1st 6383   2ndc2nd 6384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-fv 5497  df-1st 6385  df-2nd 6386
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