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Theorem xpinpreima 24296
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 3605 . 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 6360 . . . . 5  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
3 ffn 5583 . . . . 5  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
4 fncnvima2 5844 . . . . 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 5737 . . . . . 6  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  r )  =  ( 1st `  r ) )
76eleq1d 2501 . . . . 5  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 r )  e.  A  <->  ( 1st `  r
)  e.  A ) )
87rabbiia 2938 . . . 4  |-  { r  e.  ( _V  X.  _V )  |  (
( 1st  |`  ( _V 
X.  _V ) ) `  r )  e.  A }  =  { r  e.  ( _V  X.  _V )  |  ( 1st `  r )  e.  A }
95, 8eqtri 2455 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" A )  =  { r  e.  ( _V  X.  _V )  |  ( 1st `  r
)  e.  A }
10 f2ndres 6361 . . . . 5  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5583 . . . . 5  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fncnvima2 5844 . . . . 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 5737 . . . . . 6  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  r )  =  ( 2nd `  r ) )
1514eleq1d 2501 . . . . 5  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 r )  e.  B  <->  ( 2nd `  r
)  e.  B ) )
1615rabbiia 2938 . . . 4  |-  { r  e.  ( _V  X.  _V )  |  (
( 2nd  |`  ( _V 
X.  _V ) ) `  r )  e.  B }  =  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r )  e.  B }
1713, 16eqtri 2455 . . 3  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" B )  =  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r
)  e.  B }
189, 17ineq12i 3532 . 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 6376 . 2  |-  ( A  X.  B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }
201, 18, 193eqtr4ri 2466 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 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    i^i cin 3311    X. cxp 4868   `'ccnv 4869    |` cres 4872   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446   1stc1st 6339   2ndc2nd 6340
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-1st 6341  df-2nd 6342
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