Theorem unirnioo 8842

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 8817	. . . 4 |- ( -oo (,) +oo ) = RR
2		ioof 8840	. . . . . 6 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 5046	. . . . . 6 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4	2, 3	ax-mp 7	. . . . 5 |- (,) Fn ( RR* X. RR* )
5		mnfxr 8694	. . . . 5 |- -oo e. RR*
6		pnfxr 8692	. . . . 5 |- +oo e. RR*
7		fnovrn 5648	. . . . 5 |- ( ( (,) Fn ( RR* X. RR* ) /\ -oo e. RR* /\ +oo e. RR* ) -> ( -oo (,) +oo ) e. ran (,) )
8	4, 5, 6, 7	mp3an 1232	. . . 4 |- ( -oo (,) +oo ) e. ran (,)
9	1, 8	eqeltrri 2111	. . 3 |- RR e. ran (,)
10		elssuni 3608	. . 3 |- ( RR e. ran (,) -> RR C_ U. ran (,) )
11	9, 10	ax-mp 7	. 2 |- RR C_ U. ran (,)
12		frn 5052	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> ran (,) C_ ~P RR )
13	2, 12	ax-mp 7	. . 3 |- ran (,) C_ ~P RR
14		sspwuni 3739	. . 3 |- ( ran (,) C_ ~P RR <-> U. ran (,) C_ RR )
15	13, 14	mpbi 133	. 2 |- U. ran (,) C_ RR
16	11, 15	eqssi 2961	1 |- RR = U. ran (,)

Syntax hints:   = wceq 1243   e. wcel 1393   C_ wss 2917   ~Pcpw 3359   U.cuni 3580   X. cxp 4343   ran crn 4346   Fn wfn 4897   -->wf 4898  (class class class)co 5512   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059   (,)cioo 8757

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-po 4033  df-iso 4034  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-ioo 8761

This theorem is referenced by: (None)