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Theorem sxbrsigalem0 29102
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 4250 . . 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 3606 . . . 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 2422 . . . . . . . . 9  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
43rnmptss 6068 . . . . . . . 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 11420 . . . . . . . . . . 11  |- +oo  e.  RR*
6 icossre 11723 . . . . . . . . . . 11  |-  ( ( e  e.  RR  /\ +oo  e.  RR* )  ->  (
e [,) +oo )  C_  RR )
75, 6mpan2 675 . . . . . . . . . 10  |-  ( e  e.  RR  ->  (
e [,) +oo )  C_  RR )
8 xpss1 4962 . . . . . . . . . 10  |-  ( ( e [,) +oo )  C_  RR  ->  ( (
e [,) +oo )  X.  RR )  C_  ( RR  X.  RR ) )
97, 8syl 17 . . . . . . . . 9  |-  ( e  e.  RR  ->  (
( e [,) +oo )  X.  RR )  C_  ( RR  X.  RR ) )
10 ovex 6334 . . . . . . . . . . 11  |-  ( e [,) +oo )  e. 
_V
11 reex 9638 . . . . . . . . . . 11  |-  RR  e.  _V
1210, 11xpex 6610 . . . . . . . . . 10  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
1312elpw 3987 . . . . . . . . 9  |-  ( ( ( e [,) +oo )  X.  RR )  e. 
~P ( RR  X.  RR )  <->  ( ( e [,) +oo )  X.  RR )  C_  ( RR  X.  RR ) )
149, 13sylibr 215 . . . . . . . 8  |-  ( e  e.  RR  ->  (
( e [,) +oo )  X.  RR )  e. 
~P ( RR  X.  RR ) )
154, 14mprg 2785 . . . . . . 7  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) 
C_  ~P ( RR  X.  RR )
1615sseli 3460 . . . . . 6  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
1716elpwid 3991 . . . . 5  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  C_  ( RR  X.  RR ) )
18 eqid 2422 . . . . . . . . 9  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
1918rnmptss 6068 . . . . . . . 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 11723 . . . . . . . . . . 11  |-  ( ( f  e.  RR  /\ +oo  e.  RR* )  ->  (
f [,) +oo )  C_  RR )
215, 20mpan2 675 . . . . . . . . . 10  |-  ( f  e.  RR  ->  (
f [,) +oo )  C_  RR )
22 xpss2 4963 . . . . . . . . . 10  |-  ( ( f [,) +oo )  C_  RR  ->  ( RR  X.  ( f [,) +oo ) )  C_  ( RR  X.  RR ) )
2321, 22syl 17 . . . . . . . . 9  |-  ( f  e.  RR  ->  ( RR  X.  ( f [,) +oo ) )  C_  ( RR  X.  RR ) )
24 ovex 6334 . . . . . . . . . . 11  |-  ( f [,) +oo )  e. 
_V
2511, 24xpex 6610 . . . . . . . . . 10  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
2625elpw 3987 . . . . . . . . 9  |-  ( ( RR  X.  ( f [,) +oo ) )  e.  ~P ( RR 
X.  RR )  <->  ( RR  X.  ( f [,) +oo ) )  C_  ( RR  X.  RR ) )
2723, 26sylibr 215 . . . . . . . 8  |-  ( f  e.  RR  ->  ( RR  X.  ( f [,) +oo ) )  e.  ~P ( RR  X.  RR ) )
2819, 27mprg 2785 . . . . . . 7  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  C_  ~P ( RR  X.  RR )
2928sseli 3460 . . . . . 6  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
3029elpwid 3991 . . . . 5  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  -> 
z  C_  ( RR  X.  RR ) )
3117, 30jaoi 380 . . . 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 198 . . 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 2786 . 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 5637 . . . . . 6  |-  Fun  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
35 rexr 9694 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e. 
RR* )
365a1i 11 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  -> +oo  e.  RR* )
37 ltpnf 11430 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  < +oo )
38 lbico1 11697 . . . . . . . . . . 11  |-  ( ( ( 1st `  z
)  e.  RR*  /\ +oo  e.  RR*  /\  ( 1st `  z )  < +oo )  ->  ( 1st `  z
)  e.  ( ( 1st `  z ) [,) +oo ) )
3935, 36, 37, 38syl3anc 1264 . . . . . . . . . 10  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e.  ( ( 1st `  z
) [,) +oo )
)
4039anim1i 570 . . . . . . . . 9  |-  ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR )  -> 
( ( 1st `  z
)  e.  ( ( 1st `  z ) [,) +oo )  /\  ( 2nd `  z )  e.  RR ) )
4140anim2i 571 . . . . . . . 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 6841 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z
)  e.  RR ) ) )
43 elxp7 6841 . . . . . . . 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 269 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( ( 1st `  z ) [,) +oo )  X.  RR ) )
45 xp1st 6838 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
46 oveq1 6313 . . . . . . . . . 10  |-  ( e  =  ( 1st `  z
)  ->  ( e [,) +oo )  =  ( ( 1st `  z
) [,) +oo )
)
4746xpeq1d 4876 . . . . . . . . 9  |-  ( e  =  ( 1st `  z
)  ->  ( (
e [,) +oo )  X.  RR )  =  ( ( ( 1st `  z
) [,) +oo )  X.  RR ) )
48 ovex 6334 . . . . . . . . . 10  |-  ( ( 1st `  z ) [,) +oo )  e. 
_V
4948, 11xpex 6610 . . . . . . . . 9  |-  ( ( ( 1st `  z
) [,) +oo )  X.  RR )  e.  _V
5047, 3, 49fvmpt 5965 . . . . . . . 8  |-  ( ( 1st `  z )  e.  RR  ->  (
( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `  ( 1st `  z ) )  =  ( ( ( 1st `  z ) [,) +oo )  X.  RR ) )
5145, 50syl 17 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  ( ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `
 ( 1st `  z
) )  =  ( ( ( 1st `  z
) [,) +oo )  X.  RR ) )
5244, 51eleqtrrd 2510 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `  ( 1st `  z ) ) )
53 elunirn2 28253 . . . . . 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 667 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  z  e. 
U. ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) )
5554ssriv 3468 . . . 4  |-  ( RR 
X.  RR )  C_  U.
ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
56 ssun3 3631 . . . 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 4238 . . 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 3497 . 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 3480 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 369    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080    u. cun 3434    C_ wss 3436   ~Pcpw 3981   U.cuni 4219   class class class wbr 4423    |-> cmpt 4482    X. cxp 4851   ran crn 4854   Fun wfun 5595   ` cfv 5601  (class class class)co 6306   1stc1st 6806   2ndc2nd 6807   RRcr 9546   +oocpnf 9680   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-pre-lttri 9621  ax-pre-lttrn 9622
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-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-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-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-ico 11649
This theorem is referenced by:  sxbrsigalem3  29103  sxbrsigalem2  29117
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