Theorem unirnioo 10269

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 10244	. . . 4 |- ( -oo (,) +oo ) = RR
2		ioof 10267	. . . . . 6 |- (,) : ( RR* X. RR* ) --> ~P RR
3		ffn 5489	. . . . . 6 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) )
4	2, 3	ax-mp 5	. . . . 5 |- (,) Fn ( RR* X. RR* )
5		mnfxr 8295	. . . . 5 |- -oo e. RR*
6		pnfxr 8291	. . . . 5 |- +oo e. RR*
7		fnovrn 6180	. . . . 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 2305	. . 3 |- RR e. ran (,)
10		elssuni 3926	. . 3 |- ( RR e. ran (,) -> RR C_ U. ran (,) )
11	9, 10	ax-mp 5	. 2 |- RR C_ U. ran (,)
12		frn 5498	. . . 4 |- ( (,) : ( RR* X. RR* ) --> ~P RR -> ran (,) C_ ~P RR )
13	2, 12	ax-mp 5	. . 3 |- ran (,) C_ ~P RR
14		sspwuni 4060	. . 3 |- ( ran (,) C_ ~P RR <-> U. ran (,) C_ RR )
15	13, 14	mpbi 145	. 2 |- U. ran (,) C_ RR
16	11, 15	eqssi 3244	1 |- RR = U. ran (,)

Syntax hints:   = wceq 1398   e. wcel 2202   C_ wss 3201   ~Pcpw 3656   U.cuni 3898   X. cxp 4729   ran crn 4732   Fn wfn 5328   -->wf 5329  (class class class)co 6028   RRcr 8091   +oocpnf 8270   -oocmnf 8271   RR*cxr 8272   (,)cioo 10184

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-ioo 10188

This theorem is referenced by:  uniretop  15336  tgioo  15365