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Theorem xpen 6559
Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.)
Assertion
Ref Expression
xpen  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )

Proof of Theorem xpen
Dummy variables  f  g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bren 6462 . . . 4  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
21biimpi 118 . . 3  |-  ( A 
~~  B  ->  E. f 
f : A -1-1-onto-> B )
32adantr 270 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  E. f  f : A
-1-1-onto-> B )
4 bren 6462 . . . . 5  |-  ( C 
~~  D  <->  E. g 
g : C -1-1-onto-> D )
54biimpi 118 . . . 4  |-  ( C 
~~  D  ->  E. g 
g : C -1-1-onto-> D )
65ad2antlr 473 . . 3  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  f : A -1-1-onto-> B
)  ->  E. g 
g : C -1-1-onto-> D )
7 relen 6459 . . . . . . 7  |-  Rel  ~~
87brrelexi 4479 . . . . . 6  |-  ( A 
~~  B  ->  A  e.  _V )
97brrelexi 4479 . . . . . 6  |-  ( C 
~~  D  ->  C  e.  _V )
10 xpexg 4552 . . . . . 6  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  X.  C
)  e.  _V )
118, 9, 10syl2an 283 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  e.  _V )
1211ad2antrr 472 . . . 4  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  ( A  X.  C )  e. 
_V )
13 simplr 497 . . . . . 6  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  f : A -1-1-onto-> B )
14 f1ofn 5254 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  f  Fn  A )
15 dffn5im 5350 . . . . . . . 8  |-  ( f  Fn  A  ->  f  =  ( x  e.  A  |->  ( f `  x ) ) )
1614, 15syl 14 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f  =  ( x  e.  A  |->  ( f `  x
) ) )
17 f1oeq1 5244 . . . . . . 7  |-  ( f  =  ( x  e.  A  |->  ( f `  x ) )  -> 
( f : A -1-1-onto-> B  <->  ( x  e.  A  |->  ( f `  x ) ) : A -1-1-onto-> B ) )
1813, 16, 173syl 17 . . . . . 6  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  (
f : A -1-1-onto-> B  <->  ( x  e.  A  |->  ( f `
 x ) ) : A -1-1-onto-> B ) )
1913, 18mpbid 145 . . . . 5  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  (
x  e.  A  |->  ( f `  x ) ) : A -1-1-onto-> B )
20 simpr 108 . . . . . 6  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  g : C -1-1-onto-> D )
21 f1ofn 5254 . . . . . . . 8  |-  ( g : C -1-1-onto-> D  ->  g  Fn  C )
22 dffn5im 5350 . . . . . . . 8  |-  ( g  Fn  C  ->  g  =  ( y  e.  C  |->  ( g `  y ) ) )
2321, 22syl 14 . . . . . . 7  |-  ( g : C -1-1-onto-> D  ->  g  =  ( y  e.  C  |->  ( g `  y
) ) )
24 f1oeq1 5244 . . . . . . 7  |-  ( g  =  ( y  e.  C  |->  ( g `  y ) )  -> 
( g : C -1-1-onto-> D  <->  ( y  e.  C  |->  ( g `  y ) ) : C -1-1-onto-> D ) )
2520, 23, 243syl 17 . . . . . 6  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  (
g : C -1-1-onto-> D  <->  ( y  e.  C  |->  ( g `
 y ) ) : C -1-1-onto-> D ) )
2620, 25mpbid 145 . . . . 5  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  (
y  e.  C  |->  ( g `  y ) ) : C -1-1-onto-> D )
2719, 26xpf1o 6558 . . . 4  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  (
x  e.  A , 
y  e.  C  |->  <.
( f `  x
) ,  ( g `
 y ) >.
) : ( A  X.  C ) -1-1-onto-> ( B  X.  D ) )
28 f1oeng 6472 . . . 4  |-  ( ( ( A  X.  C
)  e.  _V  /\  ( x  e.  A ,  y  e.  C  |-> 
<. ( f `  x
) ,  ( g `
 y ) >.
) : ( A  X.  C ) -1-1-onto-> ( B  X.  D ) )  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
2912, 27, 28syl2anc 403 . . 3  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  ( A  X.  C )  ~~  ( B  X.  D
) )
306, 29exlimddv 1826 . 2  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  f : A -1-1-onto-> B
)  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
313, 30exlimddv 1826 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   <.cop 3449   class class class wbr 3845    |-> cmpt 3899    X. cxp 4436    Fn wfn 5010   -1-1-onto->wf1o 5014   ` cfv 5015    |-> cmpt2 5654    ~~ cen 6453
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-en 6456
This theorem is referenced by:  xpnnen  11481  xpomen  11482
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