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Theorem xpcomf1o 6687
Description: The canonical bijection from  ( A  X.  B ) to  ( B  X.  A ). (Contributed by Mario Carneiro, 23-Apr-2014.)
Hypothesis
Ref Expression
xpcomf1o.1  |-  F  =  ( x  e.  ( A  X.  B ) 
|->  U. `' { x } )
Assertion
Ref Expression
xpcomf1o  |-  F :
( A  X.  B
)
-1-1-onto-> ( B  X.  A
)
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    F( x)

Proof of Theorem xpcomf1o
StepHypRef Expression
1 relxp 4618 . . . 4  |-  Rel  ( A  X.  B )
2 cnvf1o 6090 . . . 4  |-  ( Rel  ( A  X.  B
)  ->  ( x  e.  ( A  X.  B
)  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B
) )
31, 2ax-mp 5 . . 3  |-  ( x  e.  ( A  X.  B )  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B )
4 xpcomf1o.1 . . . 4  |-  F  =  ( x  e.  ( A  X.  B ) 
|->  U. `' { x } )
5 f1oeq1 5326 . . . 4  |-  ( F  =  ( x  e.  ( A  X.  B
)  |->  U. `' { x } )  ->  ( F : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B )  <->  ( x  e.  ( A  X.  B
)  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B
) ) )
64, 5ax-mp 5 . . 3  |-  ( F : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B )  <->  ( x  e.  ( A  X.  B
)  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B
) )
73, 6mpbir 145 . 2  |-  F :
( A  X.  B
)
-1-1-onto-> `' ( A  X.  B )
8 cnvxp 4927 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
9 f1oeq3 5328 . . 3  |-  ( `' ( A  X.  B
)  =  ( B  X.  A )  -> 
( F : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B
)  <->  F : ( A  X.  B ) -1-1-onto-> ( B  X.  A ) ) )
108, 9ax-mp 5 . 2  |-  ( F : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B )  <->  F :
( A  X.  B
)
-1-1-onto-> ( B  X.  A
) )
117, 10mpbi 144 1  |-  F :
( A  X.  B
)
-1-1-onto-> ( B  X.  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1316   {csn 3497   U.cuni 3706    |-> cmpt 3959    X. cxp 4507   `'ccnv 4508   Rel wrel 4514   -1-1-onto->wf1o 5092
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-1st 6006  df-2nd 6007
This theorem is referenced by:  xpcomco  6688  xpcomen  6689
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