Theorem unirnioo 10392

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

Step	Hyp	Ref	Expression
1		ioomax 10367	. . . 4 |- ( -oo (,) +oo ) = RR
2		ioof 10390	. . . . . 6 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 5536	. . . . . 6 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4	2, 3	ax-mp 5	. . . . 5 |- (,) Fn ( RR* X. RR* )
5		mnfxr 8388	. . . . 5 |- -oo e. RR*
6		pnfxr 8384	. . . . 5 |- +oo e. RR*
7		fnovrn 6240	. . . . 5 |- ( ( (,) Fn ( RR* X. RR* ) /\ -oo e. RR* /\ +oo e. RR* ) -> ( -oo (,) +oo ) e. ran (,) )
8	4, 5, 6, 7	mp3an 1378	. . . 4 |- ( -oo (,) +oo ) e. ran (,)
9	1, 8	eqeltrri 2312	. . 3 |- RR e. ran (,)
10		elssuni 3963	. . 3 |- ( RR e. ran (,) -> RR C_ U. ran (,) )
11	9, 10	ax-mp 5	. 2 |- RR C_ U. ran (,)
12		frn 5545	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> ran (,) C_ ~P RR )
13	2, 12	ax-mp 5	. . 3 |- ran (,) C_ ~P RR
14		sspwuni 4097	. . 3 |- ( ran (,) C_ ~P RR <-> U. ran (,) C_ RR )
15	13, 14	mpbi 145	. 2 |- U. ran (,) C_ RR
16	11, 15	eqssi 3264	1 |- RR = U. ran (,)

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3688   U.cuni 3935   X. cxp 4774   ran crn 4777   Fn wfn 5375   -->wf 5376  (class class class)co 6088   RRcr 8184   +oocpnf 8363   -oocmnf 8364   RR*cxr 8365   (,)cioo 10307

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-ioo 10311

This theorem is used by:  uniretop  15751  tgioo  15780