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Theorem xpcomf1o 7618
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 5116 . . . 4  |-  Rel  ( A  X.  B )
2 cnvf1o 6894 . . . 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 5813 . . . 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 5430 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
9 f1oeq3 5815 . . 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 1379   {csn 4033   U.cuni 4251    |-> cmpt 4511    X. cxp 5003   `'ccnv 5004   Rel wrel 5010   -1-1-onto->wf1o 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1st 6795  df-2nd 6796
This theorem is referenced by:  xpcomco  7619  xpcomen  7620  omf1o  7632
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