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Theorem xpen 6705
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 6607 . . . 4  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
21biimpi 119 . . 3  |-  ( A 
~~  B  ->  E. f 
f : A -1-1-onto-> B )
32adantr 272 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  E. f  f : A
-1-1-onto-> B )
4 bren 6607 . . . . 5  |-  ( C 
~~  D  <->  E. g 
g : C -1-1-onto-> D )
54biimpi 119 . . . 4  |-  ( C 
~~  D  ->  E. g 
g : C -1-1-onto-> D )
65ad2antlr 478 . . 3  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  f : A -1-1-onto-> B
)  ->  E. g 
g : C -1-1-onto-> D )
7 relen 6604 . . . . . . 7  |-  Rel  ~~
87brrelex1i 4550 . . . . . 6  |-  ( A 
~~  B  ->  A  e.  _V )
97brrelex1i 4550 . . . . . 6  |-  ( C 
~~  D  ->  C  e.  _V )
10 xpexg 4621 . . . . . 6  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  X.  C
)  e.  _V )
118, 9, 10syl2an 285 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  e.  _V )
1211ad2antrr 477 . . . 4  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  ( A  X.  C )  e. 
_V )
13 simplr 502 . . . . . 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 5334 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  f  Fn  A )
15 dffn5im 5433 . . . . . . . 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 5324 . . . . . . 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 146 . . . . 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 109 . . . . . 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 5334 . . . . . . . 8  |-  ( g : C -1-1-onto-> D  ->  g  Fn  C )
22 dffn5im 5433 . . . . . . . 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 5324 . . . . . . 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 146 . . . . 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 6704 . . . 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 6617 . . . 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 406 . . 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 1852 . 2  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  f : A -1-1-onto-> B
)  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
313, 30exlimddv 1852 1  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  ~~  ( B  X.  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   _Vcvv 2658   <.cop 3498   class class class wbr 3897    |-> cmpt 3957    X. cxp 4505    Fn wfn 5086   -1-1-onto->wf1o 5090   ` cfv 5091    e. cmpo 5742    ~~ cen 6598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-en 6601
This theorem is referenced by:  xpdjuen  7038  xpnnen  11802  xpomen  11803  qnnen  11839
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