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Theorem xpcomf1o 6621
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 4576 . . . 4  |-  Rel  ( A  X.  B )
2 cnvf1o 6028 . . . 4  |-  ( Rel  ( A  X.  B
)  ->  ( x  e.  ( A  X.  B
)  |->  U. `' { x } ) : ( A  X.  B ) -1-1-onto-> `' ( A  X.  B
) )
31, 2ax-mp 7 . . 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 5279 . . . 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 7 . . 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 4883 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
9 f1oeq3 5281 . . 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 7 . 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 1296   {csn 3466   U.cuni 3675    |-> cmpt 3921    X. cxp 4465   `'ccnv 4466   Rel wrel 4472   -1-1-onto->wf1o 5048
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-1st 5949  df-2nd 5950
This theorem is referenced by:  xpcomco  6622  xpcomen  6623
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