Theorem icoreunrn 33207

Description: The union of all closed-below, open-above intervals of reals is the set of reals. (Contributed by ML, 27-Jul-2020.)

Hypothesis

Ref	Expression
icoreunrn.1	|- I = ( [,) " ( RR X. RR ) )

Assertion

Ref	Expression
icoreunrn	|- RR = U. I

Proof of Theorem icoreunrn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexr 10085	. . . . . . 7 |- ( x e. RR -> x e. RR* )
2		peano2re 10209	. . . . . . . 8 |- ( x e. RR -> ( x + 1 ) e. RR )
3		rexr 10085	. . . . . . . 8 |- ( ( x + 1 ) e. RR -> ( x + 1 ) e. RR* )
4	2, 3	syl 17	. . . . . . 7 |- ( x e. RR -> ( x + 1 ) e. RR* )
5		ltp1 10861	. . . . . . 7 |- ( x e. RR -> x < ( x + 1 ) )
6		lbico1 12228	. . . . . . 7 |- ( ( x e. RR* /\ ( x + 1 ) e. RR* /\ x < ( x + 1 ) ) -> x e. ( x [,) ( x + 1 ) ) )
7	1, 4, 5, 6	syl3anc 1326	. . . . . 6 |- ( x e. RR -> x e. ( x [,) ( x + 1 ) ) )
8		df-ov 6653	. . . . . 6 |- ( x [,) ( x + 1 ) ) = ( [,) ` <. x , ( x + 1 ) >. )
9	7, 8	syl6eleq 2711	. . . . 5 |- ( x e. RR -> x e. ( [,) ` <. x , ( x + 1 ) >. ) )
10		opelxpi 5148	. . . . . . 7 |- ( ( x e. RR /\ ( x + 1 ) e. RR ) -> <. x , ( x + 1 ) >. e. ( RR X. RR ) )
11	2, 10	mpdan 702	. . . . . 6 |- ( x e. RR -> <. x , ( x + 1 ) >. e. ( RR X. RR ) )
12		fvres 6207	. . . . . 6 |- ( <. x , ( x + 1 ) >. e. ( RR X. RR ) -> ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) = ( [,) ` <. x , ( x + 1 ) >. ) )
13	11, 12	syl 17	. . . . 5 |- ( x e. RR -> ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) = ( [,) ` <. x , ( x + 1 ) >. ) )
14	9, 13	eleqtrrd 2704	. . . 4 |- ( x e. RR -> x e. ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) )
15		icoreresf 33200	. . . . . . . 8 |- ( [,) |` ( RR X. RR ) ) : ( RR X. RR ) --> ~P RR
16	15	fdmi 6052	. . . . . . 7 |- dom ( [,) |` ( RR X. RR ) ) = ( RR X. RR )
17	10, 16	syl6eleqr 2712	. . . . . 6 |- ( ( x e. RR /\ ( x + 1 ) e. RR ) -> <. x , ( x + 1 ) >. e. dom ( [,) |` ( RR X. RR ) ) )
18	2, 17	mpdan 702	. . . . 5 |- ( x e. RR -> <. x , ( x + 1 ) >. e. dom ( [,) |` ( RR X. RR ) ) )
19		ffun 6048	. . . . . . . 8 |- ( ( [,) |` ( RR X. RR ) ) : ( RR X. RR ) --> ~P RR -> Fun ( [,) |` ( RR X. RR ) ) )
20	15, 19	ax-mp 5	. . . . . . 7 |- Fun ( [,) |` ( RR X. RR ) )
21		fvelrn 6352	. . . . . . 7 |- ( ( Fun ( [,) |` ( RR X. RR ) ) /\ <. x , ( x + 1 ) >. e. dom ( [,) |` ( RR X. RR ) ) ) -> ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) e. ran ( [,) |` ( RR X. RR ) ) )
22	20, 21	mpan 706	. . . . . 6 |- ( <. x , ( x + 1 ) >. e. dom ( [,) |` ( RR X. RR ) ) -> ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) e. ran ( [,) |` ( RR X. RR ) ) )
23		icoreunrn.1	. . . . . . 7 |- I = ( [,) " ( RR X. RR ) )
24		df-ima 5127	. . . . . . 7 |- ( [,) " ( RR X. RR ) ) = ran ( [,) |` ( RR X. RR ) )
25	23, 24	eqtri 2644	. . . . . 6 |- I = ran ( [,) |` ( RR X. RR ) )
26	22, 25	syl6eleqr 2712	. . . . 5 |- ( <. x , ( x + 1 ) >. e. dom ( [,) |` ( RR X. RR ) ) -> ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) e. I )
27	18, 26	syl 17	. . . 4 |- ( x e. RR -> ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) e. I )
28		elunii 4441	. . . 4 |- ( ( x e. ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) /\ ( ( [,) |` ( RR X. RR ) ) ` <. x , ( x + 1 ) >. ) e. I ) -> x e. U. I )
29	14, 27, 28	syl2anc 693	. . 3 |- ( x e. RR -> x e. U. I )
30	29	ssriv 3607	. 2 |- RR C_ U. I
31		frn 6053	. . . . 5 |- ( ( [,) |` ( RR X. RR ) ) : ( RR X. RR ) --> ~P RR -> ran ( [,) |` ( RR X. RR ) ) C_ ~P RR )
32	15, 31	ax-mp 5	. . . 4 |- ran ( [,) |` ( RR X. RR ) ) C_ ~P RR
33	25, 32	eqsstri 3635	. . 3 |- I C_ ~P RR
34		uniss 4458	. . . 4 |- ( I C_ ~P RR -> U. I C_ U. ~P RR )
35		unipw 4918	. . . 4 |- U. ~P RR = RR
36	34, 35	syl6sseq 3651	. . 3 |- ( I C_ ~P RR -> U. I C_ RR )
37	33, 36	ax-mp 5	. 2 |- U. I C_ RR
38	30, 37	eqssi 3619	1 |- RR = U. I

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   <.cop 4183   U.cuni 4436   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   [,)cico 12177

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-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

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-reu 2919  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-riota 6611  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-sub 10268  df-neg 10269  df-ico 12181

This theorem is referenced by:  istoprelowl  33208  relowlssretop  33211  relowlpssretop  33212