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Theorem xpinpreima2 24297
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 4974 . . . . . 6  |-  ( E  X.  F )  C_  ( _V  X.  _V )
2 rabss2 3418 . . . . . 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 733 . . . . . . 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 731 . . . . . . . . 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 741 . . . . . . . . 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 3341 . . . . . . . 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 732 . . . . . . . . 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 742 . . . . . . . . 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 3341 . . . . . . . 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 519 . . . . . . 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 6371 . . . . . . 7  |-  ( r  e.  ( E  X.  F )  <->  ( r  e.  ( _V  X.  _V )  /\  ( ( 1st `  r )  e.  E  /\  ( 2nd `  r
)  e.  F ) ) )
134, 11, 12sylanbrc 646 . . . . . 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 23987 . . . . 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 3357 . . . 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 6376 . . . 4  |-  ( A  X.  B )  =  { r  e.  ( _V  X.  _V )  |  ( ( 1st `  r )  e.  A  /\  ( 2nd `  r
)  e.  B ) }
1715, 16syl6reqr 2486 . . 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 3605 . . 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 2485 . 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 6360 . . . . 5  |-  ( 1st  |`  ( E  X.  F
) ) : ( E  X.  F ) --> E
21 ffn 5583 . . . . 5  |-  ( ( 1st  |`  ( E  X.  F ) ) : ( E  X.  F
) --> E  ->  ( 1st  |`  ( E  X.  F ) )  Fn  ( E  X.  F
) )
22 fncnvima2 5844 . . . . 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 5737 . . . . . 6  |-  ( r  e.  ( E  X.  F )  ->  (
( 1st  |`  ( E  X.  F ) ) `
 r )  =  ( 1st `  r
) )
2524eleq1d 2501 . . . . 5  |-  ( r  e.  ( E  X.  F )  ->  (
( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A  <->  ( 1st `  r
)  e.  A ) )
2625rabbiia 2938 . . . 4  |-  { r  e.  ( E  X.  F )  |  ( ( 1st  |`  ( E  X.  F ) ) `
 r )  e.  A }  =  {
r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }
2723, 26eqtri 2455 . . 3  |-  ( `' ( 1st  |`  ( E  X.  F ) )
" A )  =  { r  e.  ( E  X.  F )  |  ( 1st `  r
)  e.  A }
28 f2ndres 6361 . . . . 5  |-  ( 2nd  |`  ( E  X.  F
) ) : ( E  X.  F ) --> F
29 ffn 5583 . . . . 5  |-  ( ( 2nd  |`  ( E  X.  F ) ) : ( E  X.  F
) --> F  ->  ( 2nd  |`  ( E  X.  F ) )  Fn  ( E  X.  F
) )
30 fncnvima2 5844 . . . . 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 5737 . . . . . 6  |-  ( r  e.  ( E  X.  F )  ->  (
( 2nd  |`  ( E  X.  F ) ) `
 r )  =  ( 2nd `  r
) )
3332eleq1d 2501 . . . . 5  |-  ( r  e.  ( E  X.  F )  ->  (
( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B  <->  ( 2nd `  r
)  e.  B ) )
3433rabbiia 2938 . . . 4  |-  { r  e.  ( E  X.  F )  |  ( ( 2nd  |`  ( E  X.  F ) ) `
 r )  e.  B }  =  {
r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
3531, 34eqtri 2455 . . 3  |-  ( `' ( 2nd  |`  ( E  X.  F ) )
" B )  =  { r  e.  ( E  X.  F )  |  ( 2nd `  r
)  e.  B }
3627, 35ineq12i 3532 . 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 2485 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 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    i^i cin 3311    C_ wss 3312    X. cxp 4868   `'ccnv 4869    |` cres 4872   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  cnre2csqima  24301  sxbrsigalem2  24628  sxbrsiga  24632
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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