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Theorem xpinpreima 24109
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 3557 . 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 6308 . . . . 5  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
3 ffn 5532 . . . . 5  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
4 fncnvima2 5792 . . . . 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 5686 . . . . . 6  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  r )  =  ( 1st `  r ) )
76eleq1d 2454 . . . . 5  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 r )  e.  A  <->  ( 1st `  r
)  e.  A ) )
87rabbiia 2890 . . . 4  |-  { r  e.  ( _V  X.  _V )  |  (
( 1st  |`  ( _V 
X.  _V ) ) `  r )  e.  A }  =  { r  e.  ( _V  X.  _V )  |  ( 1st `  r )  e.  A }
95, 8eqtri 2408 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" A )  =  { r  e.  ( _V  X.  _V )  |  ( 1st `  r
)  e.  A }
10 f2ndres 6309 . . . . 5  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5532 . . . . 5  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fncnvima2 5792 . . . . 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 5686 . . . . . 6  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  r )  =  ( 2nd `  r ) )
1514eleq1d 2454 . . . . 5  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 r )  e.  B  <->  ( 2nd `  r
)  e.  B ) )
1615rabbiia 2890 . . . 4  |-  { r  e.  ( _V  X.  _V )  |  (
( 2nd  |`  ( _V 
X.  _V ) ) `  r )  e.  B }  =  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r )  e.  B }
1713, 16eqtri 2408 . . 3  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" B )  =  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r
)  e.  B }
189, 17ineq12i 3484 . 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 6324 . 2  |-  ( A  X.  B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }
201, 18, 193eqtr4ri 2419 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 359    = wceq 1649    e. wcel 1717   {crab 2654   _Vcvv 2900    i^i cin 3263    X. cxp 4817   `'ccnv 4818    |` cres 4821   "cima 4822    Fn wfn 5390   -->wf 5391   ` cfv 5395   1stc1st 6287   2ndc2nd 6288
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-1st 6289  df-2nd 6290
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