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Theorem xpinpreima2 23293
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 4795 . . . . . . 7  |-  ( E  X.  F )  C_  ( _V  X.  _V )
2 rabss2 3258 . . . . . . 7  |-  ( ( 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, 2ax-mp 8 . . . . . 6  |-  { 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
) }
43a1i 10 . . . . 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
) } )
5 simprl 732 . . . . . . . 8  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  r  e.  ( _V  X.  _V ) )
6 simpll 730 . . . . . . . . . 10  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  A  C_  E )
7 simprrl 740 . . . . . . . . . 10  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  ( 1st `  r )  e.  A )
86, 7sseldd 3183 . . . . . . . . 9  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  ( 1st `  r )  e.  E )
9 simplr 731 . . . . . . . . . 10  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  B  C_  F )
10 simprrr 741 . . . . . . . . . 10  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  ( 2nd `  r )  e.  B )
119, 10sseldd 3183 . . . . . . . . 9  |-  ( ( ( A  C_  E  /\  B  C_  F )  /\  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) ) )  ->  ( 2nd `  r )  e.  F )
128, 11jca 518 . . . . . . . 8  |-  ( ( ( 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 ) )
135, 12jca 518 . . . . . . 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 )  /\  (
( 1st `  r
)  e.  E  /\  ( 2nd `  r )  e.  F ) ) )
14 elxp7 6154 . . . . . . 7  |-  ( r  e.  ( E  X.  F )  <->  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  E  /\  ( 2nd `  r
)  e.  F ) ) )
1513, 14sylibr 203 . . . . . 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
) )
1615rabss3d 23138 . . . . 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 ) } )
174, 16eqssd 3198 . . . 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 ) } )
18 xp2 6159 . . . 4  |-  ( A  X.  B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }
1917, 18syl6reqr 2336 . . 3  |-  ( ( A  C_  E  /\  B  C_  F )  -> 
( A  X.  B
)  =  { r  e.  ( E  X.  F )  |  ( ( 1st `  r
)  e.  A  /\  ( 2nd `  r )  e.  B ) } )
20 inrab 3442 . . 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 ) }
2119, 20syl6eqr 2335 . 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 }
) )
22 f1stres 6143 . . . . 5  |-  ( 1st  |`  ( E  X.  F
) ) : ( E  X.  F ) --> E
23 ffn 5391 . . . . 5  |-  ( ( 1st  |`  ( E  X.  F ) ) : ( E  X.  F
) --> E  ->  ( 1st  |`  ( E  X.  F ) )  Fn  ( E  X.  F
) )
24 fncnvima2 5649 . . . . 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 } )
2522, 23, 24mp2b 9 . . . 4  |-  ( `' ( 1st  |`  ( E  X.  F ) )
" A )  =  { r  e.  ( E  X.  F )  |  ( ( 1st  |`  ( E  X.  F
) ) `  r
)  e.  A }
26 fvres 5544 . . . . . 6  |-  ( r  e.  ( E  X.  F )  ->  (
( 1st  |`  ( E  X.  F ) ) `
 r )  =  ( 1st `  r
) )
2726eleq1d 2351 . . . . 5  |-  ( r  e.  ( E  X.  F )  ->  (
( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A  <->  ( 1st `  r
)  e.  A ) )
2827rabbiia 2780 . . . 4  |-  { r  e.  ( E  X.  F )  |  ( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A }  =  {
r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }
2925, 28eqtri 2305 . . 3  |-  ( `' ( 1st  |`  ( E  X.  F ) )
" A )  =  { r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }
30 f2ndres 6144 . . . . 5  |-  ( 2nd  |`  ( E  X.  F
) ) : ( E  X.  F ) --> F
31 ffn 5391 . . . . 5  |-  ( ( 2nd  |`  ( E  X.  F ) ) : ( E  X.  F
) --> F  ->  ( 2nd  |`  ( E  X.  F ) )  Fn  ( E  X.  F
) )
32 fncnvima2 5649 . . . . 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 } )
3330, 31, 32mp2b 9 . . . 4  |-  ( `' ( 2nd  |`  ( E  X.  F ) )
" B )  =  { r  e.  ( E  X.  F )  |  ( ( 2nd  |`  ( E  X.  F
) ) `  r
)  e.  B }
34 fvres 5544 . . . . . 6  |-  ( r  e.  ( E  X.  F )  ->  (
( 2nd  |`  ( E  X.  F ) ) `
 r )  =  ( 2nd `  r
) )
3534eleq1d 2351 . . . . 5  |-  ( r  e.  ( E  X.  F )  ->  (
( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B  <->  ( 2nd `  r
)  e.  B ) )
3635rabbiia 2780 . . . 4  |-  { r  e.  ( E  X.  F )  |  ( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B }  =  {
r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
3733, 36eqtri 2305 . . 3  |-  ( `' ( 2nd  |`  ( E  X.  F ) )
" B )  =  { r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
3829, 37ineq12i 3370 . 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 }
)
3921, 38syl6eqr 2335 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   {crab 2549   _Vcvv 2790    i^i cin 3153    C_ wss 3154    X. cxp 4689   `'ccnv 4690    |` cres 4693   "cima 4694    Fn wfn 5252   -->wf 5253   ` cfv 5257   1stc1st 6122   2ndc2nd 6123
This theorem is referenced by:  cnre2csqima  23297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265  df-1st 6124  df-2nd 6125
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