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Theorem xpen 6407
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 6315 . . . 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 6315 . . . . 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 6312 . . . . . . 7  |-  Rel  ~~
87brrelexi 4430 . . . . . 6  |-  ( A 
~~  B  ->  A  e.  _V )
97brrelexi 4430 . . . . . 6  |-  ( C 
~~  D  ->  C  e.  _V )
10 xpexg 4500 . . . . . 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 5178 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  f  Fn  A )
15 dffn5im 5271 . . . . . . . 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 5168 . . . . . . 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 5178 . . . . . . . 8  |-  ( g : C -1-1-onto-> D  ->  g  Fn  C )
22 dffn5im 5271 . . . . . . . 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 5168 . . . . . . 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 6406 . . . 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 6325 . . . 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 1821 . 2  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  f : A -1-1-onto-> B
)  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
313, 30exlimddv 1821 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 1285   E.wex 1422    e. wcel 1434   _Vcvv 2610   <.cop 3419   class class class wbr 3805    |-> cmpt 3859    X. cxp 4389    Fn wfn 4947   -1-1-onto->wf1o 4951   ` cfv 4952    |-> cmpt2 5565    ~~ cen 6306
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-en 6309
This theorem is referenced by:  xpnnen  10814  xpomen  10815
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