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Theorem unirnioo 9947
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 9922 . . . 4  |-  ( -oo (,) +oo )  =  RR
2 ioof 9945 . . . . . 6  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 ffn 5360 . . . . . 6  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  (,)  Fn  ( RR*  X.  RR* )
)
42, 3ax-mp 5 . . . . 5  |-  (,)  Fn  ( RR*  X.  RR* )
5 mnfxr 7991 . . . . 5  |- -oo  e.  RR*
6 pnfxr 7987 . . . . 5  |- +oo  e.  RR*
7 fnovrn 6015 . . . . 5  |-  ( ( (,)  Fn  ( RR*  X. 
RR* )  /\ -oo  e.  RR*  /\ +oo  e.  RR* )  ->  ( -oo (,) +oo )  e.  ran  (,) )
84, 5, 6, 7mp3an 1337 . . . 4  |-  ( -oo (,) +oo )  e.  ran  (,)
91, 8eqeltrri 2251 . . 3  |-  RR  e.  ran  (,)
10 elssuni 3835 . . 3  |-  ( RR  e.  ran  (,)  ->  RR  C_  U. ran  (,) )
119, 10ax-mp 5 . 2  |-  RR  C_  U.
ran  (,)
12 frn 5369 . . . 4  |-  ( (,)
: ( RR*  X.  RR* )
--> ~P RR  ->  ran  (,)  C_  ~P RR )
132, 12ax-mp 5 . . 3  |-  ran  (,)  C_ 
~P RR
14 sspwuni 3968 . . 3  |-  ( ran 
(,)  C_  ~P RR  <->  U. ran  (,)  C_  RR )
1513, 14mpbi 145 . 2  |-  U. ran  (,)  C_  RR
1611, 15eqssi 3171 1  |-  RR  =  U. ran  (,)
Colors of variables: wff set class
Syntax hints:    = wceq 1353    e. wcel 2148    C_ wss 3129   ~Pcpw 3574   U.cuni 3807    X. cxp 4620   ran crn 4623    Fn wfn 5206   -->wf 5207  (class class class)co 5868   RRcr 7788   +oocpnf 7966   -oocmnf 7967   RR*cxr 7968   (,)cioo 9862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-po 4292  df-iso 4293  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-ioo 9866
This theorem is referenced by:  uniretop  13658  tgioo  13679
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