Users' Mathboxes Mathbox for ML < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  icoreunrn Structured version   Unicode version

Theorem icoreunrn 31727
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.
StepHypRef Expression
1 rexr 9694 . . . . . . 7  |-  ( x  e.  RR  ->  x  e.  RR* )
2 peano2re 9814 . . . . . . . 8  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR )
3 rexr 9694 . . . . . . . 8  |-  ( ( x  +  1 )  e.  RR  ->  (
x  +  1 )  e.  RR* )
42, 3syl 17 . . . . . . 7  |-  ( x  e.  RR  ->  (
x  +  1 )  e.  RR* )
5 ltp1 10451 . . . . . . 7  |-  ( x  e.  RR  ->  x  <  ( x  +  1 ) )
6 lbico1 11697 . . . . . . 7  |-  ( ( x  e.  RR*  /\  (
x  +  1 )  e.  RR*  /\  x  <  ( x  +  1 ) )  ->  x  e.  ( x [,) (
x  +  1 ) ) )
71, 4, 5, 6syl3anc 1264 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  ( x [,) (
x  +  1 ) ) )
8 df-ov 6309 . . . . . 6  |-  ( x [,) ( x  + 
1 ) )  =  ( [,) `  <. x ,  ( x  + 
1 ) >. )
97, 8syl6eleq 2517 . . . . 5  |-  ( x  e.  RR  ->  x  e.  ( [,) `  <. x ,  ( x  + 
1 ) >. )
)
10 opelxpi 4885 . . . . . . 7  |-  ( ( x  e.  RR  /\  ( x  +  1
)  e.  RR )  ->  <. x ,  ( x  +  1 )
>.  e.  ( RR  X.  RR ) )
112, 10mpdan 672 . . . . . 6  |-  ( x  e.  RR  ->  <. x ,  ( x  + 
1 ) >.  e.  ( RR  X.  RR ) )
12 fvres 5896 . . . . . 6  |-  ( <.
x ,  ( x  +  1 ) >.  e.  ( RR  X.  RR )  ->  ( ( [,)  |`  ( RR  X.  RR ) ) `  <. x ,  ( x  + 
1 ) >. )  =  ( [,) `  <. x ,  ( x  + 
1 ) >. )
)
1311, 12syl 17 . . . . 5  |-  ( x  e.  RR  ->  (
( [,)  |`  ( RR 
X.  RR ) ) `
 <. x ,  ( x  +  1 )
>. )  =  ( [,) `  <. x ,  ( x  +  1 )
>. ) )
149, 13eleqtrrd 2510 . . . 4  |-  ( x  e.  RR  ->  x  e.  ( ( [,)  |`  ( RR  X.  RR ) ) `
 <. x ,  ( x  +  1 )
>. ) )
15 icoreresf 31720 . . . . . . . 8  |-  ( [,)  |`  ( RR  X.  RR ) ) : ( RR  X.  RR ) --> ~P RR
1615fdmi 5751 . . . . . . 7  |-  dom  ( [,)  |`  ( RR  X.  RR ) )  =  ( RR  X.  RR )
1710, 16syl6eleqr 2518 . . . . . 6  |-  ( ( x  e.  RR  /\  ( x  +  1
)  e.  RR )  ->  <. x ,  ( x  +  1 )
>.  e.  dom  ( [,)  |`  ( RR  X.  RR ) ) )
182, 17mpdan 672 . . . . 5  |-  ( x  e.  RR  ->  <. x ,  ( x  + 
1 ) >.  e.  dom  ( [,)  |`  ( RR  X.  RR ) ) )
19 ffun 5748 . . . . . . . 8  |-  ( ( [,)  |`  ( RR  X.  RR ) ) : ( RR  X.  RR )
--> ~P RR  ->  Fun  ( [,)  |`  ( RR  X.  RR ) ) )
2015, 19ax-mp 5 . . . . . . 7  |-  Fun  ( [,)  |`  ( RR  X.  RR ) )
21 fvelrn 6031 . . . . . . 7  |-  ( ( Fun  ( [,)  |`  ( RR  X.  RR ) )  /\  <. x ,  ( x  +  1 )
>.  e.  dom  ( [,)  |`  ( RR  X.  RR ) ) )  -> 
( ( [,)  |`  ( RR  X.  RR ) ) `
 <. x ,  ( x  +  1 )
>. )  e.  ran  ( [,)  |`  ( RR  X.  RR ) ) )
2220, 21mpan 674 . . . . . 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 4866 . . . . . . 7  |-  ( [,) " ( RR  X.  RR ) )  =  ran  ( [,)  |`  ( RR  X.  RR ) )
2523, 24eqtri 2451 . . . . . 6  |-  I  =  ran  ( [,)  |`  ( RR  X.  RR ) )
2622, 25syl6eleqr 2518 . . . . 5  |-  ( <.
x ,  ( x  +  1 ) >.  e.  dom  ( [,)  |`  ( RR  X.  RR ) )  ->  ( ( [,)  |`  ( RR  X.  RR ) ) `  <. x ,  ( x  + 
1 ) >. )  e.  I )
2718, 26syl 17 . . . 4  |-  ( x  e.  RR  ->  (
( [,)  |`  ( RR 
X.  RR ) ) `
 <. x ,  ( x  +  1 )
>. )  e.  I
)
28 elunii 4224 . . . 4  |-  ( ( x  e.  ( ( [,)  |`  ( RR  X.  RR ) ) `  <. x ,  ( x  +  1 ) >.
)  /\  ( ( [,)  |`  ( RR  X.  RR ) ) `  <. x ,  ( x  + 
1 ) >. )  e.  I )  ->  x  e.  U. I )
2914, 27, 28syl2anc 665 . . 3  |-  ( x  e.  RR  ->  x  e.  U. I )
3029ssriv 3468 . 2  |-  RR  C_  U. I
31 frn 5752 . . . . 5  |-  ( ( [,)  |`  ( RR  X.  RR ) ) : ( RR  X.  RR )
--> ~P RR  ->  ran  ( [,)  |`  ( RR  X.  RR ) )  C_  ~P RR )
3215, 31ax-mp 5 . . . 4  |-  ran  ( [,)  |`  ( RR  X.  RR ) )  C_  ~P RR
3325, 32eqsstri 3494 . . 3  |-  I  C_  ~P RR
34 uniss 4240 . . . 4  |-  ( I 
C_  ~P RR  ->  U. I  C_ 
U. ~P RR )
35 unipw 4671 . . . 4  |-  U. ~P RR  =  RR
3634, 35syl6sseq 3510 . . 3  |-  ( I 
C_  ~P RR  ->  U. I  C_  RR )
3733, 36ax-mp 5 . 2  |-  U. I  C_  RR
3830, 37eqssi 3480 1  |-  RR  =  U. I
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3436   ~Pcpw 3981   <.cop 4004   U.cuni 4219   class class class wbr 4423    X. cxp 4851   dom cdm 4853   ran crn 4854    |` cres 4855   "cima 4856   Fun wfun 5595   -->wf 5597   ` cfv 5601  (class class class)co 6306   RRcr 9546   1c1 9548    + caddc 9550   RR*cxr 9682    < clt 9683   [,)cico 11645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-1st 6808  df-2nd 6809  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-ico 11649
This theorem is referenced by:  istoprelowl  31728  relowlssretop  31731  relowlpssretop  31732
  Copyright terms: Public domain W3C validator