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Theorem sxbrsigalem0 27882
Description: The closed half-spaces of  ( RR  X.  RR ) cover  ( RR 
X.  RR ). (Contributed by Thierry Arnoux, 11-Oct-2017.)
Assertion
Ref Expression
sxbrsigalem0  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  =  ( RR  X.  RR )
Distinct variable group:    e, f

Proof of Theorem sxbrsigalem0
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 unissb 4277 . . 3  |-  ( U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) 
C_  ( RR  X.  RR )  <->  A. z  e.  ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) z  C_  ( RR  X.  RR ) )
2 elun 3645 . . . 4  |-  ( z  e.  ( ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) ) )  <-> 
( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  \/  z  e.  ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) ) )
3 eqid 2467 . . . . . . . . 9  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
43rnmptss 6048 . . . . . . . 8  |-  ( A. e  e.  RR  (
( e [,) +oo )  X.  RR )  e. 
~P ( RR  X.  RR )  ->  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) 
C_  ~P ( RR  X.  RR ) )
5 pnfxr 11317 . . . . . . . . . . 11  |- +oo  e.  RR*
6 icossre 11601 . . . . . . . . . . 11  |-  ( ( e  e.  RR  /\ +oo  e.  RR* )  ->  (
e [,) +oo )  C_  RR )
75, 6mpan2 671 . . . . . . . . . 10  |-  ( e  e.  RR  ->  (
e [,) +oo )  C_  RR )
8 xpss1 5109 . . . . . . . . . 10  |-  ( ( e [,) +oo )  C_  RR  ->  ( (
e [,) +oo )  X.  RR )  C_  ( RR  X.  RR ) )
97, 8syl 16 . . . . . . . . 9  |-  ( e  e.  RR  ->  (
( e [,) +oo )  X.  RR )  C_  ( RR  X.  RR ) )
10 ovex 6307 . . . . . . . . . . 11  |-  ( e [,) +oo )  e. 
_V
11 reex 9579 . . . . . . . . . . 11  |-  RR  e.  _V
1210, 11xpex 6711 . . . . . . . . . 10  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
1312elpw 4016 . . . . . . . . 9  |-  ( ( ( e [,) +oo )  X.  RR )  e. 
~P ( RR  X.  RR )  <->  ( ( e [,) +oo )  X.  RR )  C_  ( RR  X.  RR ) )
149, 13sylibr 212 . . . . . . . 8  |-  ( e  e.  RR  ->  (
( e [,) +oo )  X.  RR )  e. 
~P ( RR  X.  RR ) )
154, 14mprg 2827 . . . . . . 7  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) 
C_  ~P ( RR  X.  RR )
1615sseli 3500 . . . . . 6  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
1716elpwid 4020 . . . . 5  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  C_  ( RR  X.  RR ) )
18 eqid 2467 . . . . . . . . 9  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
1918rnmptss 6048 . . . . . . . 8  |-  ( A. f  e.  RR  ( RR  X.  ( f [,) +oo ) )  e.  ~P ( RR  X.  RR )  ->  ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  C_  ~P ( RR  X.  RR ) )
20 icossre 11601 . . . . . . . . . . 11  |-  ( ( f  e.  RR  /\ +oo  e.  RR* )  ->  (
f [,) +oo )  C_  RR )
215, 20mpan2 671 . . . . . . . . . 10  |-  ( f  e.  RR  ->  (
f [,) +oo )  C_  RR )
22 xpss2 5110 . . . . . . . . . 10  |-  ( ( f [,) +oo )  C_  RR  ->  ( RR  X.  ( f [,) +oo ) )  C_  ( RR  X.  RR ) )
2321, 22syl 16 . . . . . . . . 9  |-  ( f  e.  RR  ->  ( RR  X.  ( f [,) +oo ) )  C_  ( RR  X.  RR ) )
24 ovex 6307 . . . . . . . . . . 11  |-  ( f [,) +oo )  e. 
_V
2511, 24xpex 6711 . . . . . . . . . 10  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
2625elpw 4016 . . . . . . . . 9  |-  ( ( RR  X.  ( f [,) +oo ) )  e.  ~P ( RR 
X.  RR )  <->  ( RR  X.  ( f [,) +oo ) )  C_  ( RR  X.  RR ) )
2723, 26sylibr 212 . . . . . . . 8  |-  ( f  e.  RR  ->  ( RR  X.  ( f [,) +oo ) )  e.  ~P ( RR  X.  RR ) )
2819, 27mprg 2827 . . . . . . 7  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  C_  ~P ( RR  X.  RR )
2928sseli 3500 . . . . . 6  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
3029elpwid 4020 . . . . 5  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  -> 
z  C_  ( RR  X.  RR ) )
3117, 30jaoi 379 . . . 4  |-  ( ( z  e.  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  \/  z  e.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  -> 
z  C_  ( RR  X.  RR ) )
322, 31sylbi 195 . . 3  |-  ( z  e.  ( ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) ) )  ->  z  C_  ( RR  X.  RR ) )
331, 32mprgbir 2828 . 2  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  C_  ( RR  X.  RR )
34 funmpt 5622 . . . . . 6  |-  Fun  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
35 rexr 9635 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e. 
RR* )
365a1i 11 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  -> +oo  e.  RR* )
37 ltpnf 11327 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  < +oo )
38 lbico1 11575 . . . . . . . . . . 11  |-  ( ( ( 1st `  z
)  e.  RR*  /\ +oo  e.  RR*  /\  ( 1st `  z )  < +oo )  ->  ( 1st `  z
)  e.  ( ( 1st `  z ) [,) +oo ) )
3935, 36, 37, 38syl3anc 1228 . . . . . . . . . 10  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e.  ( ( 1st `  z
) [,) +oo )
)
4039anim1i 568 . . . . . . . . 9  |-  ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR )  -> 
( ( 1st `  z
)  e.  ( ( 1st `  z ) [,) +oo )  /\  ( 2nd `  z )  e.  RR ) )
4140anim2i 569 . . . . . . . 8  |-  ( ( z  e.  ( _V 
X.  _V )  /\  (
( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR ) )  ->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  ( ( 1st `  z
) [,) +oo )  /\  ( 2nd `  z
)  e.  RR ) ) )
42 elxp7 6814 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z
)  e.  RR ) ) )
43 elxp7 6814 . . . . . . . 8  |-  ( z  e.  ( ( ( 1st `  z ) [,) +oo )  X.  RR )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  ( ( 1st `  z
) [,) +oo )  /\  ( 2nd `  z
)  e.  RR ) ) )
4441, 42, 433imtr4i 266 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( ( 1st `  z ) [,) +oo )  X.  RR ) )
45 xp1st 6811 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
46 oveq1 6289 . . . . . . . . . 10  |-  ( e  =  ( 1st `  z
)  ->  ( e [,) +oo )  =  ( ( 1st `  z
) [,) +oo )
)
4746xpeq1d 5022 . . . . . . . . 9  |-  ( e  =  ( 1st `  z
)  ->  ( (
e [,) +oo )  X.  RR )  =  ( ( ( 1st `  z
) [,) +oo )  X.  RR ) )
48 ovex 6307 . . . . . . . . . 10  |-  ( ( 1st `  z ) [,) +oo )  e. 
_V
4948, 11xpex 6711 . . . . . . . . 9  |-  ( ( ( 1st `  z
) [,) +oo )  X.  RR )  e.  _V
5047, 3, 49fvmpt 5948 . . . . . . . 8  |-  ( ( 1st `  z )  e.  RR  ->  (
( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `  ( 1st `  z ) )  =  ( ( ( 1st `  z ) [,) +oo )  X.  RR ) )
5145, 50syl 16 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `
 ( 1st `  z
) )  =  ( ( ( 1st `  z
) [,) +oo )  X.  RR ) )
5244, 51eleqtrrd 2558 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `  ( 1st `  z ) ) )
53 elunirn2 27161 . . . . . 6  |-  ( ( Fun  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  /\  z  e.  ( (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `
 ( 1st `  z
) ) )  -> 
z  e.  U. ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) )
5434, 52, 53sylancr 663 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  z  e. 
U. ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) )
5554ssriv 3508 . . . 4  |-  ( RR 
X.  RR )  C_  U.
ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
56 ssun3 3669 . . . 4  |-  ( ( RR  X.  RR ) 
C_  U. ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
( RR  X.  RR )  C_  ( U. ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  U. ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) ) )
5755, 56ax-mp 5 . . 3  |-  ( RR 
X.  RR )  C_  ( U. ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
U. ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )
58 uniun 4264 . . 3  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  =  ( U. ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  U. ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )
5957, 58sseqtr4i 3537 . 2  |-  ( RR 
X.  RR )  C_  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u. 
ran  ( f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) ) )
6033, 59eqssi 3520 1  |-  U. ( ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  u.  ran  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) ) )  =  ( RR  X.  RR )
Colors of variables: wff setvar class
Syntax hints:    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ran crn 5000   Fun wfun 5580   ` cfv 5586  (class class class)co 6282   1stc1st 6779   2ndc2nd 6780   RRcr 9487   +oocpnf 9621   RR*cxr 9623    < clt 9624   [,)cico 11527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttri 9562  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-ico 11531
This theorem is referenced by:  sxbrsigalem3  27883  sxbrsigalem2  27897
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