Theorem unirnioo 10306

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 10281	. . . 4 |- ( -oo (,) +oo ) = RR
2		ioof 10304	. . . . . 6 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 5508	. . . . . 6 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4	2, 3	ax-mp 5	. . . . 5 |- (,) Fn ( RR* X. RR* )
5		mnfxr 8330	. . . . 5 |- -oo e. RR*
6		pnfxr 8326	. . . . 5 |- +oo e. RR*
7		fnovrn 6202	. . . . 5 |- ( ( (,) Fn ( RR* X. RR* ) /\ -oo e. RR* /\ +oo e. RR* ) -> ( -oo (,) +oo ) e. ran (,) )
8	4, 5, 6, 7	mp3an 1374	. . . 4 |- ( -oo (,) +oo ) e. ran (,)
9	1, 8	eqeltrri 2306	. . 3 |- RR e. ran (,)
10		elssuni 3942	. . 3 |- ( RR e. ran (,) -> RR C_ U. ran (,) )
11	9, 10	ax-mp 5	. 2 |- RR C_ U. ran (,)
12		frn 5517	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> ran (,) C_ ~P RR )
13	2, 12	ax-mp 5	. . 3 |- ran (,) C_ ~P RR
14		sspwuni 4076	. . 3 |- ( ran (,) C_ ~P RR <-> U. ran (,) C_ RR )
15	13, 14	mpbi 145	. 2 |- U. ran (,) C_ RR
16	11, 15	eqssi 3254	1 |- RR = U. ran (,)

Syntax hints:   = wceq 1398   e. wcel 2203   C_ wss 3211   ~Pcpw 3669   U.cuni 3914   X. cxp 4747   ran crn 4750   Fn wfn 5347   -->wf 5348  (class class class)co 6050   RRcr 8126   +oocpnf 8305   -oocmnf 8306   RR*cxr 8307   (,)cioo 10221

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-ioo 10225

This theorem is referenced by:  uniretop  15390  tgioo  15419