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Theorem sxbrsigalem0 28403
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 4283 . . 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 3641 . . . 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 2457 . . . . . . . . 9  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
43rnmptss 6061 . . . . . . . 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 11346 . . . . . . . . . . 11  |- +oo  e.  RR*
6 icossre 11630 . . . . . . . . . . 11  |-  ( ( e  e.  RR  /\ +oo  e.  RR* )  ->  (
e [,) +oo )  C_  RR )
75, 6mpan2 671 . . . . . . . . . 10  |-  ( e  e.  RR  ->  (
e [,) +oo )  C_  RR )
8 xpss1 5120 . . . . . . . . . 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 6324 . . . . . . . . . . 11  |-  ( e [,) +oo )  e. 
_V
11 reex 9600 . . . . . . . . . . 11  |-  RR  e.  _V
1210, 11xpex 6603 . . . . . . . . . 10  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
1312elpw 4021 . . . . . . . . 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 2820 . . . . . . 7  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) 
C_  ~P ( RR  X.  RR )
1615sseli 3495 . . . . . 6  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
1716elpwid 4025 . . . . 5  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  C_  ( RR  X.  RR ) )
18 eqid 2457 . . . . . . . . 9  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
1918rnmptss 6061 . . . . . . . 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 11630 . . . . . . . . . . 11  |-  ( ( f  e.  RR  /\ +oo  e.  RR* )  ->  (
f [,) +oo )  C_  RR )
215, 20mpan2 671 . . . . . . . . . 10  |-  ( f  e.  RR  ->  (
f [,) +oo )  C_  RR )
22 xpss2 5121 . . . . . . . . . 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 6324 . . . . . . . . . . 11  |-  ( f [,) +oo )  e. 
_V
2511, 24xpex 6603 . . . . . . . . . 10  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
2625elpw 4021 . . . . . . . . 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 2820 . . . . . . 7  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  C_  ~P ( RR  X.  RR )
2928sseli 3495 . . . . . 6  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
3029elpwid 4025 . . . . 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 2821 . 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 5630 . . . . . 6  |-  Fun  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
35 rexr 9656 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e. 
RR* )
365a1i 11 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  -> +oo  e.  RR* )
37 ltpnf 11356 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  < +oo )
38 lbico1 11604 . . . . . . . . . . 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 6832 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z
)  e.  RR ) ) )
43 elxp7 6832 . . . . . . . 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 6829 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
46 oveq1 6303 . . . . . . . . . 10  |-  ( e  =  ( 1st `  z
)  ->  ( e [,) +oo )  =  ( ( 1st `  z
) [,) +oo )
)
4746xpeq1d 5031 . . . . . . . . 9  |-  ( e  =  ( 1st `  z
)  ->  ( (
e [,) +oo )  X.  RR )  =  ( ( ( 1st `  z
) [,) +oo )  X.  RR ) )
48 ovex 6324 . . . . . . . . . 10  |-  ( ( 1st `  z ) [,) +oo )  e. 
_V
4948, 11xpex 6603 . . . . . . . . 9  |-  ( ( ( 1st `  z
) [,) +oo )  X.  RR )  e.  _V
5047, 3, 49fvmpt 5956 . . . . . . . 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 2548 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `  ( 1st `  z ) ) )
53 elunirn2 27632 . . . . . 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 3503 . . . 4  |-  ( RR 
X.  RR )  C_  U.
ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
56 ssun3 3665 . . . 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 4270 . . 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 3532 . 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 3515 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 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469    C_ wss 3471   ~Pcpw 4015   U.cuni 4251   class class class wbr 4456    |-> cmpt 4515    X. cxp 5006   ran crn 5009   Fun wfun 5588   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798   RRcr 9508   +oocpnf 9642   RR*cxr 9644    < clt 9645   [,)cico 11556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-ico 11560
This theorem is referenced by:  sxbrsigalem3  28404  sxbrsigalem2  28418
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