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Theorem unirnioo 10759
Description: The union of the range of the open interval function. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)
Assertion
Ref Expression
unirnioo  |-  RR  =  U. ran  (,)

Proof of Theorem unirnioo
StepHypRef Expression
1 ioomax 10740 . . . 4  |-  (  -oo (,) 
+oo )  =  RR
2 ioof 10757 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5405 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
42, 3ax-mp 8 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
5 mnfxr 10472 . . . . 5  |-  -oo  e.  RR*
6 pnfxr 10471 . . . . 5  |-  +oo  e.  RR*
7 fnovrn 6011 . . . . 5  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  -oo  e.  RR*  /\  +oo  e.  RR* )  ->  (  -oo (,) 
+oo )  e.  ran  (,) )
84, 5, 6, 7mp3an 1277 . . . 4  |-  (  -oo (,) 
+oo )  e.  ran  (,)
91, 8eqeltrri 2367 . . 3  |-  RR  e.  ran  (,)
10 elssuni 3871 . . 3  |-  ( RR  e.  ran  (,)  ->  RR  C_  U. ran  (,) )
119, 10ax-mp 8 . 2  |-  RR  C_  U.
ran  (,)
12 frn 5411 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
132, 12ax-mp 8 . . 3  |-  ran  (,)  C_ 
~P RR
14 sspwuni 4003 . . 3  |-  ( ran 
(,)  C_  ~P RR  <->  U. ran  (,)  C_  RR )
1513, 14mpbi 199 . 2  |-  U. ran  (,)  C_  RR
1611, 15eqssi 3208 1  |-  RR  =  U. ran  (,)
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696    C_ wss 3165   ~Pcpw 3638   U.cuni 3843    X. cxp 4703   ran crn 4706    Fn wfn 5266   -->wf 5267  (class class class)co 5874   RRcr 8752    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882   (,)cioo 10672
This theorem is referenced by:  pnfnei  16966  mnfnei  16967  uniretop  18287  tgioo  18318  xrtgioo  18328  bndth  18472
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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