Theorem ixxf 10279

Description: The set of intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 16-Nov-2013.)

Hypothesis

Ref	Expression
ixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )

Assertion

Ref	Expression
ixxf	|- O : ( RR* X. RR* ) --> ~P RR*

Distinct variable groups:   x, y, z, R   x, S, y, z

Allowed substitution hints:   O( x, y, z)

Proof of Theorem ixxf

Step	Hyp	Ref	Expression
1		ssrab2 3333	. . . 4 |- { z e. RR* | ( x R z /\ z S y ) } C_ RR*
2		xrex 10237	. . . . 5 |- RR* e. _V
3	2	elpw2 4288	. . . 4 |- ( { z e. RR* | ( x R z /\ z S y ) } e. ~P RR* <-> { z e. RR* | ( x R z /\ z S y ) } C_ RR* )
4	1, 3	mpbir 146	. . 3 |- { z e. RR* | ( x R z /\ z S y ) } e. ~P RR*
5	4	rgen2w 2606	. 2 |- A. x e. RR* A. y e. RR* { z e. RR* | ( x R z /\ z S y ) } e. ~P RR*
6		ixx.1	. . 3 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
7	6	fmpo 6427	. 2 |- ( A. x e. RR* A. y e. RR* { z e. RR* | ( x R z /\ z S y ) } e. ~P RR* <-> O : ( RR* X. RR* ) --> ~P RR* )
8	5, 7	mpbi 145	1 |- O : ( RR* X. RR* ) --> ~P RR*

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   X. cxp 4767   -->wf 5368   e. cmpo 6077   RR*cxr 8349

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354

This theorem is referenced by:  ixxex  10280  ixxssxr  10281  iccf  10353