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Theorem sxbrsigalem0 24574
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 4005 . . 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 3448 . . . 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 2404 . . . . . . . . 9  |-  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) 
+oo )  X.  RR ) )
43rnmptss 5856 . . . . . . . 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 10669 . . . . . . . . . . 11  |-  +oo  e.  RR*
6 icossre 10947 . . . . . . . . . . 11  |-  ( ( e  e.  RR  /\  +oo 
e.  RR* )  ->  (
e [,)  +oo )  C_  RR )
75, 6mpan2 653 . . . . . . . . . 10  |-  ( e  e.  RR  ->  (
e [,)  +oo )  C_  RR )
8 xpss1 4943 . . . . . . . . . 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 6065 . . . . . . . . . . 11  |-  ( e [,)  +oo )  e.  _V
11 reex 9037 . . . . . . . . . . 11  |-  RR  e.  _V
1210, 11xpex 4949 . . . . . . . . . 10  |-  ( ( e [,)  +oo )  X.  RR )  e.  _V
1312elpw 3765 . . . . . . . . 9  |-  ( ( ( e [,)  +oo )  X.  RR )  e. 
~P ( RR  X.  RR )  <->  ( ( e [,)  +oo )  X.  RR )  C_  ( RR  X.  RR ) )
149, 13sylibr 204 . . . . . . . 8  |-  ( e  e.  RR  ->  (
( e [,)  +oo )  X.  RR )  e. 
~P ( RR  X.  RR ) )
154, 14mprg 2735 . . . . . . 7  |-  ran  (
e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) 
C_  ~P ( RR  X.  RR )
1615sseli 3304 . . . . . 6  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
1716elpwid 3768 . . . . 5  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )  -> 
z  C_  ( RR  X.  RR ) )
18 eqid 2404 . . . . . . . . 9  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) )
1918rnmptss 5856 . . . . . . . 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 10947 . . . . . . . . . . 11  |-  ( ( f  e.  RR  /\  +oo 
e.  RR* )  ->  (
f [,)  +oo )  C_  RR )
215, 20mpan2 653 . . . . . . . . . 10  |-  ( f  e.  RR  ->  (
f [,)  +oo )  C_  RR )
22 xpss2 4944 . . . . . . . . . 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 6065 . . . . . . . . . . 11  |-  ( f [,)  +oo )  e.  _V
2511, 24xpex 4949 . . . . . . . . . 10  |-  ( RR 
X.  ( f [,) 
+oo ) )  e. 
_V
2625elpw 3765 . . . . . . . . 9  |-  ( ( RR  X.  ( f [,)  +oo ) )  e. 
~P ( RR  X.  RR )  <->  ( RR  X.  ( f [,)  +oo ) )  C_  ( RR  X.  RR ) )
2723, 26sylibr 204 . . . . . . . 8  |-  ( f  e.  RR  ->  ( RR  X.  ( f [,) 
+oo ) )  e. 
~P ( RR  X.  RR ) )
2819, 27mprg 2735 . . . . . . 7  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,)  +oo ) ) ) 
C_  ~P ( RR  X.  RR )
2928sseli 3304 . . . . . 6  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  ->  z  e.  ~P ( RR  X.  RR ) )
3029elpwid 3768 . . . . 5  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) 
+oo ) ) )  ->  z  C_  ( RR  X.  RR ) )
3117, 30jaoi 369 . . . 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 188 . . 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 2736 . 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 5448 . . . . . 6  |-  Fun  (
e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )
35 rexr 9086 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e. 
RR* )
365a1i 11 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  +oo  e.  RR* )
37 ltpnf 10677 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  <  +oo )
38 lbico1 10922 . . . . . . . . . . 11  |-  ( ( ( 1st `  z
)  e.  RR*  /\  +oo  e.  RR*  /\  ( 1st `  z )  <  +oo )  ->  ( 1st `  z
)  e.  ( ( 1st `  z ) [,)  +oo ) )
3935, 36, 37, 38syl3anc 1184 . . . . . . . . . 10  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e.  ( ( 1st `  z
) [,)  +oo ) )
4039anim1i 552 . . . . . . . . 9  |-  ( ( ( 1st `  z
)  e.  RR  /\  ( 2nd `  z )  e.  RR )  -> 
( ( 1st `  z
)  e.  ( ( 1st `  z ) [,)  +oo )  /\  ( 2nd `  z )  e.  RR ) )
4140anim2i 553 . . . . . . . 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 6338 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z
)  e.  RR ) ) )
43 elxp7 6338 . . . . . . . 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 258 . . . . . . 7  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( ( 1st `  z ) [,)  +oo )  X.  RR ) )
45 xp1st 6335 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
46 oveq1 6047 . . . . . . . . . 10  |-  ( e  =  ( 1st `  z
)  ->  ( e [,)  +oo )  =  ( ( 1st `  z
) [,)  +oo ) )
4746xpeq1d 4860 . . . . . . . . 9  |-  ( e  =  ( 1st `  z
)  ->  ( (
e [,)  +oo )  X.  RR )  =  ( ( ( 1st `  z
) [,)  +oo )  X.  RR ) )
48 ovex 6065 . . . . . . . . . 10  |-  ( ( 1st `  z ) [,)  +oo )  e.  _V
4948, 11xpex 4949 . . . . . . . . 9  |-  ( ( ( 1st `  z
) [,)  +oo )  X.  RR )  e.  _V
5047, 3, 49fvmpt 5765 . . . . . . . 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 2481 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) `  ( 1st `  z ) ) )
53 elunirn2 24016 . . . . . 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 645 . . . . 5  |-  ( z  e.  ( RR  X.  RR )  ->  z  e. 
U. ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) ) )
5554ssriv 3312 . . . 4  |-  ( RR 
X.  RR )  C_  U.
ran  ( e  e.  RR  |->  ( ( e [,)  +oo )  X.  RR ) )
56 ssun3 3472 . . . 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 8 . . 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 3994 . . 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 3341 . 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 3324 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 set class
Syntax hints:    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   class class class wbr 4172    e. cmpt 4226    X. cxp 4835   ran crn 4838   Fun wfun 5407   ` cfv 5413  (class class class)co 6040   1stc1st 6306   2ndc2nd 6307   RRcr 8945    +oocpnf 9073   RR*cxr 9075    < clt 9076   [,)cico 10874
This theorem is referenced by:  sxbrsigalem3  24575  sxbrsigalem2  24589
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-pre-lttri 9020  ax-pre-lttrn 9021
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-ico 10878
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