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Theorem unirnioo 11492
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 11473 . . . 4  |-  ( -oo (,) +oo )  =  RR
2 ioof 11490 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5659 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
42, 3ax-mp 5 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
5 mnfxr 11197 . . . . 5  |- -oo  e.  RR*
6 pnfxr 11195 . . . . 5  |- +oo  e.  RR*
7 fnovrn 6340 . . . . 5  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\ -oo  e.  RR*  /\ +oo  e.  RR* )  ->  ( -oo (,) +oo )  e.  ran  (,) )
84, 5, 6, 7mp3an 1315 . . . 4  |-  ( -oo (,) +oo )  e.  ran  (,)
91, 8eqeltrri 2536 . . 3  |-  RR  e.  ran  (,)
10 elssuni 4221 . . 3  |-  ( RR  e.  ran  (,)  ->  RR  C_  U. ran  (,) )
119, 10ax-mp 5 . 2  |-  RR  C_  U.
ran  (,)
12 frn 5665 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
132, 12ax-mp 5 . . 3  |-  ran  (,)  C_ 
~P RR
14 sspwuni 4356 . . 3  |-  ( ran 
(,)  C_  ~P RR  <->  U. ran  (,)  C_  RR )
1513, 14mpbi 208 . 2  |-  U. ran  (,)  C_  RR
1611, 15eqssi 3472 1  |-  RR  =  U. ran  (,)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758    C_ wss 3428   ~Pcpw 3960   U.cuni 4191    X. cxp 4938   ran crn 4941    Fn wfn 5513   -->wf 5514  (class class class)co 6192   RRcr 9384   +oocpnf 9518   -oocmnf 9519   RR*cxr 9520   (,)cioo 11403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-pre-lttri 9459  ax-pre-lttrn 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-po 4741  df-so 4742  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-1st 6679  df-2nd 6680  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-ioo 11407
This theorem is referenced by:  pnfnei  18942  mnfnei  18943  uniretop  20459  tgioo  20491  xrtgioo  20501  bndth  20648  mblfinlem3  28570  mblfinlem4  28571  ismblfin  28572
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