Theorem xpcomf1o 7008

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 4835	. . . 4 |- Rel ( A X. B )
2		cnvf1o 6389	. . . 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 5571	. . . 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 5155	. . 3 |- `' ( A X. B ) = ( B X. A )
9		f1oeq3 5573	. . 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 1397   {csn 3669   U.cuni 3893   |-> cmpt 4150   X. cxp 4723   `'ccnv 4724   Rel wrel 4730   -1-1-onto->wf1o 5325

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by:  xpcomco  7009  xpcomen  7010