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Theorem unirnioo 10996
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 10977 . . . 4  |-  (  -oo (,) 
+oo )  =  RR
2 ioof 10994 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5583 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
42, 3ax-mp 8 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
5 mnfxr 10706 . . . . 5  |-  -oo  e.  RR*
6 pnfxr 10705 . . . . 5  |-  +oo  e.  RR*
7 fnovrn 6213 . . . . 5  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\  -oo  e.  RR*  /\  +oo  e.  RR* )  ->  (  -oo (,) 
+oo )  e.  ran  (,) )
84, 5, 6, 7mp3an 1279 . . . 4  |-  (  -oo (,) 
+oo )  e.  ran  (,)
91, 8eqeltrri 2506 . . 3  |-  RR  e.  ran  (,)
10 elssuni 4035 . . 3  |-  ( RR  e.  ran  (,)  ->  RR  C_  U. ran  (,) )
119, 10ax-mp 8 . 2  |-  RR  C_  U.
ran  (,)
12 frn 5589 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
132, 12ax-mp 8 . . 3  |-  ran  (,)  C_ 
~P RR
14 sspwuni 4168 . . 3  |-  ( ran 
(,)  C_  ~P RR  <->  U. ran  (,)  C_  RR )
1513, 14mpbi 200 . 2  |-  U. ran  (,)  C_  RR
1611, 15eqssi 3356 1  |-  RR  =  U. ran  (,)
Colors of variables: wff set class
Syntax hints:    = wceq 1652    e. wcel 1725    C_ wss 3312   ~Pcpw 3791   U.cuni 4007    X. cxp 4868   ran crn 4871    Fn wfn 5441   -->wf 5442  (class class class)co 6073   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111   (,)cioo 10908
This theorem is referenced by:  pnfnei  17276  mnfnei  17277  uniretop  18788  tgioo  18819  xrtgioo  18829  bndth  18975  mblfinlem2  26235  mblfinlem3  26236  ismblfin  26237
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-pre-lttri 9056  ax-pre-lttrn 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-ioo 10912
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