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Theorem xpinpreima2 28552
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
xpinpreima2  |-  ( ( A  C_  E  /\  B  C_  F )  -> 
( A  X.  B
)  =  ( ( `' ( 1st  |`  ( E  X.  F ) )
" A )  i^i  ( `' ( 2nd  |`  ( E  X.  F
) ) " B
) ) )

Proof of Theorem xpinpreima2
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 xpss 4961 . . . . . 6  |-  ( E  X.  F )  C_  ( _V  X.  _V )
2 rabss2 3550 . . . . . 6  |-  ( ( E  X.  F ) 
C_  ( _V  X.  _V )  ->  { r  e.  ( E  X.  F )  |  ( ( 1st `  r
)  e.  A  /\  ( 2nd `  r )  e.  B ) } 
C_  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r )  e.  B
) } )
31, 2mp1i 13 . . . . 5  |-  ( ( A  C_  E  /\  B  C_  F )  ->  { r  e.  ( E  X.  F )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }  C_  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r )  e.  B
) } )
4 simprl 762 . . . . . . 7  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  r  e.  ( _V  X.  _V ) )
5 simpll 758 . . . . . . . . 9  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  A  C_  E )
6 simprrl 772 . . . . . . . . 9  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  ( 1st `  r )  e.  A )
75, 6sseldd 3471 . . . . . . . 8  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  ( 1st `  r )  e.  E )
8 simplr 760 . . . . . . . . 9  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  B  C_  F )
9 simprrr 773 . . . . . . . . 9  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  ( 2nd `  r )  e.  B )
108, 9sseldd 3471 . . . . . . . 8  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  ( 2nd `  r )  e.  F )
117, 10jca 534 . . . . . . 7  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  (
( 1st `  r
)  e.  E  /\  ( 2nd `  r )  e.  F ) )
12 elxp7 6840 . . . . . . 7  |-  ( r  e.  ( E  X.  F )  <->  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  E  /\  ( 2nd `  r
)  e.  F ) ) )
134, 11, 12sylanbrc 668 . . . . . 6  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  r  e.  ( E  X.  F
) )
1413rabss3d 27984 . . . . 5  |-  ( ( A  C_  E  /\  B  C_  F )  ->  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }  C_  { r  e.  ( E  X.  F
)  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r )  e.  B ) } )
153, 14eqssd 3487 . . . 4  |-  ( ( A  C_  E  /\  B  C_  F )  ->  { r  e.  ( E  X.  F )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }  =  { r  e.  ( _V  X.  _V )  |  (
( 1st `  r
)  e.  A  /\  ( 2nd `  r )  e.  B ) } )
16 xp2 6842 . . . 4  |-  ( A  X.  B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }
1715, 16syl6reqr 2489 . . 3  |-  ( ( A  C_  E  /\  B  C_  F )  -> 
( A  X.  B
)  =  { r  e.  ( E  X.  F )  |  ( ( 1st `  r
)  e.  A  /\  ( 2nd `  r )  e.  B ) } )
18 inrab 3751 . . 3  |-  ( { r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }  i^i  { r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
)  =  { r  e.  ( E  X.  F )  |  ( ( 1st `  r
)  e.  A  /\  ( 2nd `  r )  e.  B ) }
1917, 18syl6eqr 2488 . 2  |-  ( ( A  C_  E  /\  B  C_  F )  -> 
( A  X.  B
)  =  ( { r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }  i^i  { r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
) )
20 f1stres 6829 . . . . 5  |-  ( 1st  |`  ( E  X.  F
) ) : ( E  X.  F ) --> E
21 ffn 5746 . . . . 5  |-  ( ( 1st  |`  ( E  X.  F ) ) : ( E  X.  F
) --> E  ->  ( 1st  |`  ( E  X.  F ) )  Fn  ( E  X.  F
) )
22 fncnvima2 6019 . . . . 5  |-  ( ( 1st  |`  ( E  X.  F ) )  Fn  ( E  X.  F
)  ->  ( `' ( 1st  |`  ( E  X.  F ) ) " A )  =  {
r  e.  ( E  X.  F )  |  ( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A } )
2320, 21, 22mp2b 10 . . . 4  |-  ( `' ( 1st  |`  ( E  X.  F ) )
" A )  =  { r  e.  ( E  X.  F )  |  ( ( 1st  |`  ( E  X.  F
) ) `  r
)  e.  A }
24 fvres 5895 . . . . . 6  |-  ( r  e.  ( E  X.  F )  ->  (
( 1st  |`  ( E  X.  F ) ) `
 r )  =  ( 1st `  r
) )
2524eleq1d 2498 . . . . 5  |-  ( r  e.  ( E  X.  F )  ->  (
( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A  <->  ( 1st `  r
)  e.  A ) )
2625rabbiia 3076 . . . 4  |-  { r  e.  ( E  X.  F )  |  ( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A }  =  {
r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }
2723, 26eqtri 2458 . . 3  |-  ( `' ( 1st  |`  ( E  X.  F ) )
" A )  =  { r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }
28 f2ndres 6830 . . . . 5  |-  ( 2nd  |`  ( E  X.  F
) ) : ( E  X.  F ) --> F
29 ffn 5746 . . . . 5  |-  ( ( 2nd  |`  ( E  X.  F ) ) : ( E  X.  F
) --> F  ->  ( 2nd  |`  ( E  X.  F ) )  Fn  ( E  X.  F
) )
30 fncnvima2 6019 . . . . 5  |-  ( ( 2nd  |`  ( E  X.  F ) )  Fn  ( E  X.  F
)  ->  ( `' ( 2nd  |`  ( E  X.  F ) ) " B )  =  {
r  e.  ( E  X.  F )  |  ( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B } )
3128, 29, 30mp2b 10 . . . 4  |-  ( `' ( 2nd  |`  ( E  X.  F ) )
" B )  =  { r  e.  ( E  X.  F )  |  ( ( 2nd  |`  ( E  X.  F
) ) `  r
)  e.  B }
32 fvres 5895 . . . . . 6  |-  ( r  e.  ( E  X.  F )  ->  (
( 2nd  |`  ( E  X.  F ) ) `
 r )  =  ( 2nd `  r
) )
3332eleq1d 2498 . . . . 5  |-  ( r  e.  ( E  X.  F )  ->  (
( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B  <->  ( 2nd `  r
)  e.  B ) )
3433rabbiia 3076 . . . 4  |-  { r  e.  ( E  X.  F )  |  ( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B }  =  {
r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
3531, 34eqtri 2458 . . 3  |-  ( `' ( 2nd  |`  ( E  X.  F ) )
" B )  =  { r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
3627, 35ineq12i 3668 . 2  |-  ( ( `' ( 1st  |`  ( E  X.  F ) )
" A )  i^i  ( `' ( 2nd  |`  ( E  X.  F
) ) " B
) )  =  ( { r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }  i^i  { r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
)
3719, 36syl6eqr 2488 1  |-  ( ( A  C_  E  /\  B  C_  F )  -> 
( A  X.  B
)  =  ( ( `' ( 1st  |`  ( E  X.  F ) )
" A )  i^i  ( `' ( 2nd  |`  ( E  X.  F
) ) " B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087    i^i cin 3441    C_ wss 3442    X. cxp 4852   `'ccnv 4853    |` cres 4856   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601   1stc1st 6805   2ndc2nd 6806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-1st 6807  df-2nd 6808
This theorem is referenced by:  cnre2csqima  28556  sxbrsigalem2  28947  sxbrsiga  28951
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