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Theorem unirnioo 10743
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 10724 . . . 4  |-  (  -oo (,) 
+oo )  =  RR
2 ioof 10741 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5389 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
42, 3ax-mp 8 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
5 mnfxr 10456 . . . . 5  |-  -oo  e.  RR*
6 pnfxr 10455 . . . . 5  |-  +oo  e.  RR*
7 fnovrn 5995 . . . . 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 2354 . . 3  |-  RR  e.  ran  (,)
10 elssuni 3855 . . 3  |-  ( RR  e.  ran  (,)  ->  RR  C_  U. ran  (,) )
119, 10ax-mp 8 . 2  |-  RR  C_  U.
ran  (,)
12 frn 5395 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
132, 12ax-mp 8 . . 3  |-  ran  (,)  C_ 
~P RR
14 sspwuni 3987 . . 3  |-  ( ran 
(,)  C_  ~P RR  <->  U. ran  (,)  C_  RR )
1513, 14mpbi 199 . 2  |-  U. ran  (,)  C_  RR
1611, 15eqssi 3195 1  |-  RR  =  U. ran  (,)
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684    C_ wss 3152   ~Pcpw 3625   U.cuni 3827    X. cxp 4687   ran crn 4690    Fn wfn 5250   -->wf 5251  (class class class)co 5858   RRcr 8736    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866   (,)cioo 10656
This theorem is referenced by:  pnfnei  16950  mnfnei  16951  uniretop  18271  tgioo  18302  xrtgioo  18312  bndth  18456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-ioo 10660
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