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Theorem sxbrsigalem0 26622
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 4120 . . 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 3494 . . . 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 2441 . . . . . . . . 9  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
43rnmptss 5869 . . . . . . . 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 11088 . . . . . . . . . . 11  |- +oo  e.  RR*
6 icossre 11372 . . . . . . . . . . 11  |-  ( ( e  e.  RR  /\ +oo  e.  RR* )  ->  (
e [,) +oo )  C_  RR )
75, 6mpan2 666 . . . . . . . . . 10  |-  ( e  e.  RR  ->  (
e [,) +oo )  C_  RR )
8 xpss1 4944 . . . . . . . . . 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 6115 . . . . . . . . . . 11  |-  ( e [,) +oo )  e. 
_V
11 reex 9369 . . . . . . . . . . 11  |-  RR  e.  _V
1210, 11xpex 6507 . . . . . . . . . 10  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
1312elpw 3863 . . . . . . . . 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 2783 . . . . . . 7  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) 
C_  ~P ( RR  X.  RR )
1615sseli 3349 . . . . . 6  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
1716elpwid 3867 . . . . 5  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  C_  ( RR  X.  RR ) )
18 eqid 2441 . . . . . . . . 9  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
1918rnmptss 5869 . . . . . . . 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 11372 . . . . . . . . . . 11  |-  ( ( f  e.  RR  /\ +oo  e.  RR* )  ->  (
f [,) +oo )  C_  RR )
215, 20mpan2 666 . . . . . . . . . 10  |-  ( f  e.  RR  ->  (
f [,) +oo )  C_  RR )
22 xpss2 4945 . . . . . . . . . 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 6115 . . . . . . . . . . 11  |-  ( f [,) +oo )  e. 
_V
2511, 24xpex 6507 . . . . . . . . . 10  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
2625elpw 3863 . . . . . . . . 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 2783 . . . . . . 7  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  C_  ~P ( RR  X.  RR )
2928sseli 3349 . . . . . 6  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
3029elpwid 3867 . . . . 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 2784 . 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 5451 . . . . . 6  |-  Fun  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
35 rexr 9425 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e. 
RR* )
365a1i 11 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  -> +oo  e.  RR* )
37 ltpnf 11098 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  < +oo )
38 lbico1 11346 . . . . . . . . . . 11  |-  ( ( ( 1st `  z
)  e.  RR*  /\ +oo  e.  RR*  /\  ( 1st `  z )  < +oo )  ->  ( 1st `  z
)  e.  ( ( 1st `  z ) [,) +oo ) )
3935, 36, 37, 38syl3anc 1213 . . . . . . . . . 10  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e.  ( ( 1st `  z
) [,) +oo )
)
4039anim1i 565 . . . . . . . . 9  |-  ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR )  -> 
( ( 1st `  z
)  e.  ( ( 1st `  z ) [,) +oo )  /\  ( 2nd `  z )  e.  RR ) )
4140anim2i 566 . . . . . . . 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 6608 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z
)  e.  RR ) ) )
43 elxp7 6608 . . . . . . . 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 6605 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
46 oveq1 6097 . . . . . . . . . 10  |-  ( e  =  ( 1st `  z
)  ->  ( e [,) +oo )  =  ( ( 1st `  z
) [,) +oo )
)
4746xpeq1d 4859 . . . . . . . . 9  |-  ( e  =  ( 1st `  z
)  ->  ( (
e [,) +oo )  X.  RR )  =  ( ( ( 1st `  z
) [,) +oo )  X.  RR ) )
48 ovex 6115 . . . . . . . . . 10  |-  ( ( 1st `  z ) [,) +oo )  e. 
_V
4948, 11xpex 6507 . . . . . . . . 9  |-  ( ( ( 1st `  z
) [,) +oo )  X.  RR )  e.  _V
5047, 3, 49fvmpt 5771 . . . . . . . 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 2518 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `  ( 1st `  z ) ) )
53 elunirn2 25901 . . . . . 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 658 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  z  e. 
U. ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) )
5554ssriv 3357 . . . 4  |-  ( RR 
X.  RR )  C_  U.
ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
56 ssun3 3518 . . . 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 4107 . . 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 3386 . 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 3369 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 1364    e. wcel 1761   _Vcvv 2970    u. cun 3323    C_ wss 3325   ~Pcpw 3857   U.cuni 4088   class class class wbr 4289    e. cmpt 4347    X. cxp 4834   ran crn 4837   Fun wfun 5409   ` cfv 5415  (class class class)co 6090   1stc1st 6574   2ndc2nd 6575   RRcr 9277   +oocpnf 9411   RR*cxr 9413    < clt 9414   [,)cico 11298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-pre-lttri 9352  ax-pre-lttrn 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-ico 11302
This theorem is referenced by:  sxbrsigalem3  26623  sxbrsigalem2  26637
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