Theorem xpcomf1o 6945

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 4802	. . . 4 |- Rel ( A X. B )
2		cnvf1o 6334	. . . 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 5532	. . . 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 5120	. . 3 |- `' ( A X. B ) = ( B X. A )
9		f1oeq3 5534	. . 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 1373   {csn 3643   U.cuni 3864   |-> cmpt 4121   X. cxp 4691   `'ccnv 4692   Rel wrel 4698   -1-1-onto->wf1o 5289

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1st 6249  df-2nd 6250

This theorem is referenced by:  xpcomco  6946  xpcomen  6947