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Theorem intrn 25702
Description: Condition for an interval to belong to the range of  (,) (Contributed by FL, 5-Jan-2009.)
Assertion
Ref Expression
intrn  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e. 
ran  (,) )

Proof of Theorem intrn
StepHypRef Expression
1 ioof 10757 . 2  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
2 ffn 5405 . 2  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
3 fnovrn 6011 . . 3  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  A  e.  RR*  /\  B  e. 
RR* )  ->  ( A (,) B )  e. 
ran  (,) )
433expib 1154 . 2  |-  ( (,) 
Fn  ( RR*  X.  RR* )  ->  ( ( A  e.  RR*  /\  B  e. 
RR* )  ->  ( A (,) B )  e. 
ran  (,) ) )
51, 2, 4mp2b 9 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  e. 
ran  (,) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   ~Pcpw 3638    X. cxp 4703   ran crn 4706    Fn wfn 5266   -->wf 5267  (class class class)co 5874   RRcr 8752   RR*cxr 8882   (,)cioo 10672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-ioo 10676
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