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Theorem xpen 6823
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 6725 . . . 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 274 . 2  |-  ( ( A  ~~  B  /\  C  ~~  D )  ->  E. f  f : A
-1-1-onto-> B )
4 bren 6725 . . . . 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 486 . . 3  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  f : A -1-1-onto-> B
)  ->  E. g 
g : C -1-1-onto-> D )
7 relen 6722 . . . . . . 7  |-  Rel  ~~
87brrelex1i 4654 . . . . . 6  |-  ( A 
~~  B  ->  A  e.  _V )
97brrelex1i 4654 . . . . . 6  |-  ( C 
~~  D  ->  C  e.  _V )
10 xpexg 4725 . . . . . 6  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A  X.  C
)  e.  _V )
118, 9, 10syl2an 287 . . . . 5  |-  ( ( A  ~~  B  /\  C  ~~  D )  -> 
( A  X.  C
)  e.  _V )
1211ad2antrr 485 . . . 4  |-  ( ( ( ( A  ~~  B  /\  C  ~~  D
)  /\  f : A
-1-1-onto-> B )  /\  g : C -1-1-onto-> D )  ->  ( A  X.  C )  e. 
_V )
13 simplr 525 . . . . . 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 5443 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  f  Fn  A )
15 dffn5im 5542 . . . . . . . 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 5431 . . . . . . 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 5443 . . . . . . . 8  |-  ( g : C -1-1-onto-> D  ->  g  Fn  C )
22 dffn5im 5542 . . . . . . . 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 5431 . . . . . . 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 6822 . . . 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 6735 . . . 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 409 . . 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 1891 . 2  |-  ( ( ( A  ~~  B  /\  C  ~~  D )  /\  f : A -1-1-onto-> B
)  ->  ( A  X.  C )  ~~  ( B  X.  D ) )
313, 30exlimddv 1891 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 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730   <.cop 3586   class class class wbr 3989    |-> cmpt 4050    X. cxp 4609    Fn wfn 5193   -1-1-onto->wf1o 5197   ` cfv 5198    e. cmpo 5855    ~~ cen 6716
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-en 6719
This theorem is referenced by:  xpdjuen  7195  xpnnen  12349  xpomen  12350  qnnen  12386
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