Theorem xpcomf1o 6712

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 4643	. . . 4 |- Rel ( A X. B )
2		cnvf1o 6115	. . . 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 5351	. . . 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 145	. 2 |- F : ( A X. B ) -1-1-onto-> `' ( A X. B )
8		cnvxp 4952	. . 3 |- `' ( A X. B ) = ( B X. A )
9		f1oeq3 5353	. . 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 144	1 |- F : ( A X. B ) -1-1-onto-> ( B X. A )

Syntax hints:   <-> wb 104   = wceq 1331   {csn 3522   U.cuni 3731   |-> cmpt 3984   X. cxp 4532   `'ccnv 4533   Rel wrel 4539   -1-1-onto->wf1o 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032

This theorem is referenced by:  xpcomco  6713  xpcomen  6714