Theorem ixxf 10250

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 3327	. . . 4 |- { z e. RR* | ( x R z /\ z S y ) } C_ RR*
2		xrex 10208	. . . . 5 |- RR* e. _V
3	2	elpw2 4274	. . . 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 2600	. 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 6410	. 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 1398   e. wcel 2205   A.wral 2522   {crab 2526   C_ wss 3214   ~Pcpw 3674   class class class wbr 4114   X. cxp 4752   -->wf 5353   e. cmpo 6060   RR*cxr 8323

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  ax-cnex 8234  ax-resscn 8235

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-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  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-iun 3998  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-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328

This theorem is referenced by:  ixxex  10251  ixxssxr  10252  iccf  10324