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Theorem xpinpreima 27552
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 3770 . 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 6806 . . . . 5  |-  ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
3 ffn 5731 . . . . 5  |-  ( ( 1st  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 1st  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
4 fncnvima2 6003 . . . . 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 5880 . . . . . 6  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( 1st  |`  ( _V  X.  _V ) ) `  r )  =  ( 1st `  r ) )
76eleq1d 2536 . . . . 5  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( ( 1st  |`  ( _V  X.  _V ) ) `
 r )  e.  A  <->  ( 1st `  r
)  e.  A ) )
87rabbiia 3102 . . . 4  |-  { r  e.  ( _V  X.  _V )  |  (
( 1st  |`  ( _V 
X.  _V ) ) `  r )  e.  A }  =  { r  e.  ( _V  X.  _V )  |  ( 1st `  r )  e.  A }
95, 8eqtri 2496 . . 3  |-  ( `' ( 1st  |`  ( _V  X.  _V ) )
" A )  =  { r  e.  ( _V  X.  _V )  |  ( 1st `  r
)  e.  A }
10 f2ndres 6807 . . . . 5  |-  ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V ) --> _V
11 ffn 5731 . . . . 5  |-  ( ( 2nd  |`  ( _V  X.  _V ) ) : ( _V  X.  _V )
--> _V  ->  ( 2nd  |`  ( _V  X.  _V ) )  Fn  ( _V  X.  _V ) )
12 fncnvima2 6003 . . . . 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 5880 . . . . . 6  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( 2nd  |`  ( _V  X.  _V ) ) `  r )  =  ( 2nd `  r ) )
1514eleq1d 2536 . . . . 5  |-  ( r  e.  ( _V  X.  _V )  ->  ( ( ( 2nd  |`  ( _V  X.  _V ) ) `
 r )  e.  B  <->  ( 2nd `  r
)  e.  B ) )
1615rabbiia 3102 . . . 4  |-  { r  e.  ( _V  X.  _V )  |  (
( 2nd  |`  ( _V 
X.  _V ) ) `  r )  e.  B }  =  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r )  e.  B }
1713, 16eqtri 2496 . . 3  |-  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" B )  =  { r  e.  ( _V  X.  _V )  |  ( 2nd `  r
)  e.  B }
189, 17ineq12i 3698 . 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 6819 . 2  |-  ( A  X.  B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }
201, 18, 193eqtr4ri 2507 1  |-  ( A  X.  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) )
" B ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    i^i cin 3475    X. cxp 4997   `'ccnv 4998    |` cres 5001   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588   1stc1st 6782   2ndc2nd 6783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-1st 6784  df-2nd 6785
This theorem is referenced by: (None)
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