Theorem unirnioo 12273

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 12248	. . . 4 |- ( -oo (,) +oo ) = RR
2		ioof 12271	. . . . . 6 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 6045	. . . . . 6 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4	2, 3	ax-mp 5	. . . . 5 |- (,) Fn ( RR* X. RR* )
5		mnfxr 10096	. . . . 5 |- -oo e. RR*
6		pnfxr 10092	. . . . 5 |- +oo e. RR*
7		fnovrn 6809	. . . . 5 |- ( ( (,) Fn ( RR* X. RR* ) /\ -oo e. RR* /\ +oo e. RR* ) -> ( -oo (,) +oo ) e. ran (,) )
8	4, 5, 6, 7	mp3an 1424	. . . 4 |- ( -oo (,) +oo ) e. ran (,)
9	1, 8	eqeltrri 2698	. . 3 |- RR e. ran (,)
10		elssuni 4467	. . 3 |- ( RR e. ran (,) -> RR C_ U. ran (,) )
11	9, 10	ax-mp 5	. 2 |- RR C_ U. ran (,)
12		frn 6053	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> ran (,) C_ ~P RR )
13	2, 12	ax-mp 5	. . 3 |- ran (,) C_ ~P RR
14		sspwuni 4611	. . 3 |- ( ran (,) C_ ~P RR <-> U. ran (,) C_ RR )
15	13, 14	mpbi 220	. 2 |- U. ran (,) C_ RR
16	11, 15	eqssi 3619	1 |- RR = U. ran (,)

Syntax hints:   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   X. cxp 5112   ran crn 5115   Fn wfn 5883   -->wf 5884  (class class class)co 6650   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   (,)cioo 12175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioo 12179

This theorem is referenced by:  pnfnei  21024  mnfnei  21025  uniretop  22566  tgioo  22599  xrtgioo  22609  bndth  22757  relowlssretop  33211  relowlpssretop  33212  mblfinlem3  33448  mblfinlem4  33449  ismblfin  33450