Theorem xpcomf1o 7089

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

Step	Hyp	Ref	Expression
1		relxp 4864	. . . 4 |- Rel ( A X. B )
2		cnvf1o 6434	. . . 4 |- ( Rel ( A X. B ) -> ( x e. ( A X. B ) |-> U. `' { x } ) : ( A X. B ) -1-1-onto-> `' ( A X. B ) )
3	1, 2	ax-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 5607	. . . 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 ) ) )
6	4, 5	ax-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 ) )
7	3, 6	mpbir 146	. 2 |- F : ( A X. B ) -1-1-onto-> `' ( A X. B )
8		cnvxp 5186	. . 3 |- `' ( A X. B ) = ( B X. A )
9		f1oeq3 5609	. . 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 ) ) )
10	8, 9	ax-mp 5	. 2 |- ( F : ( A X. B ) -1-1-onto-> `' ( A X. B ) <-> F : ( A X. B ) -1-1-onto-> ( B X. A ) )
11	7, 10	mpbi 145	1 |- F : ( A X. B ) -1-1-onto-> ( B X. A )

Syntax hints:   <-> wb 105   = wceq 1398   {csn 3694   U.cuni 3919   |-> cmpt 4176   X. cxp 4752   `'ccnv 4753   Rel wrel 4759   -1-1-onto->wf1o 5356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348

This theorem is referenced by:  xpcomco  7090  xpcomen  7091