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Theorem xpcomf1o 7686
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 4960 . . . 4  |-  Rel  ( A  X.  B )
2 cnvf1o 6921 . . . 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 5827 . . . 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 214 . 2  |-  F :
( A  X.  B
)
-1-1-onto-> `' ( A  X.  B )
8 cnvxp 5272 . . 3  |-  `' ( A  X.  B )  =  ( B  X.  A )
9 f1oeq3 5829 . . 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 213 1  |-  F :
( A  X.  B
)
-1-1-onto-> ( B  X.  A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1454   {csn 3979   U.cuni 4211    |-> cmpt 4474    X. cxp 4850   `'ccnv 4851   Rel wrel 4857   -1-1-onto->wf1o 5599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-1st 6819  df-2nd 6820
This theorem is referenced by:  xpcomco  7687  xpcomen  7688  omf1o  7700
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