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Theorem xpcomf1o 7400
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 4947 . . . 4  |-  Rel  ( A  X.  B )
2 cnvf1o 6671 . . . 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 5632 . . . 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 209 . 2  |-  F :
( A  X.  B
)
-1-1-onto-> `' ( A  X.  B )
8 cnvxp 5255 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
9 f1oeq3 5634 . . 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 208 1  |-  F :
( A  X.  B
)
-1-1-onto-> ( B  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1369   {csn 3877   U.cuni 4091    e. cmpt 4350    X. cxp 4838   `'ccnv 4839   Rel wrel 4845   -1-1-onto->wf1o 5417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-1st 6577  df-2nd 6578
This theorem is referenced by:  xpcomco  7401  xpcomen  7402  omf1o  7414
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