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Theorem xpinpreima2 26336
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 4945 . . . . . 6  |-  ( E  X.  F )  C_  ( _V  X.  _V )
2 rabss2 3434 . . . . . 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 12 . . . . 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 755 . . . . . . 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 753 . . . . . . . . 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 763 . . . . . . . . 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 3356 . . . . . . . 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 754 . . . . . . . . 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 764 . . . . . . . . 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 3356 . . . . . . . 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 532 . . . . . . 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 6608 . . . . . . 7  |-  ( r  e.  ( E  X.  F )  <->  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  E  /\  ( 2nd `  r
)  e.  F ) ) )
134, 11, 12sylanbrc 664 . . . . . 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 25894 . . . . 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 3372 . . . 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 6610 . . . 4  |-  ( A  X.  B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }
1715, 16syl6reqr 2493 . . 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 3621 . . 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 2492 . 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 6597 . . . . 5  |-  ( 1st  |`  ( E  X.  F
) ) : ( E  X.  F ) --> E
21 ffn 5558 . . . . 5  |-  ( ( 1st  |`  ( E  X.  F ) ) : ( E  X.  F
) --> E  ->  ( 1st  |`  ( E  X.  F ) )  Fn  ( E  X.  F
) )
22 fncnvima2 5824 . . . . 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 5703 . . . . . 6  |-  ( r  e.  ( E  X.  F )  ->  (
( 1st  |`  ( E  X.  F ) ) `
 r )  =  ( 1st `  r
) )
2524eleq1d 2508 . . . . 5  |-  ( r  e.  ( E  X.  F )  ->  (
( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A  <->  ( 1st `  r
)  e.  A ) )
2625rabbiia 2960 . . . 4  |-  { r  e.  ( E  X.  F )  |  ( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A }  =  {
r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }
2723, 26eqtri 2462 . . 3  |-  ( `' ( 1st  |`  ( E  X.  F ) )
" A )  =  { r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }
28 f2ndres 6598 . . . . 5  |-  ( 2nd  |`  ( E  X.  F
) ) : ( E  X.  F ) --> F
29 ffn 5558 . . . . 5  |-  ( ( 2nd  |`  ( E  X.  F ) ) : ( E  X.  F
) --> F  ->  ( 2nd  |`  ( E  X.  F ) )  Fn  ( E  X.  F
) )
30 fncnvima2 5824 . . . . 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 5703 . . . . . 6  |-  ( r  e.  ( E  X.  F )  ->  (
( 2nd  |`  ( E  X.  F ) ) `
 r )  =  ( 2nd `  r
) )
3332eleq1d 2508 . . . . 5  |-  ( r  e.  ( E  X.  F )  ->  (
( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B  <->  ( 2nd `  r
)  e.  B ) )
3433rabbiia 2960 . . . 4  |-  { r  e.  ( E  X.  F )  |  ( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B }  =  {
r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
3531, 34eqtri 2462 . . 3  |-  ( `' ( 2nd  |`  ( E  X.  F ) )
" B )  =  { r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
3627, 35ineq12i 3549 . 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 2492 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 369    = wceq 1369    e. wcel 1756   {crab 2718   _Vcvv 2971    i^i cin 3326    C_ wss 3327    X. cxp 4837   `'ccnv 4838    |` cres 4841   "cima 4842    Fn wfn 5412   -->wf 5413   ` cfv 5417   1stc1st 6574   2ndc2nd 6575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-fv 5425  df-1st 6576  df-2nd 6577
This theorem is referenced by:  cnre2csqima  26340  sxbrsigalem2  26700  sxbrsiga  26704
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