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Theorem sxbrsigalem0 26701
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 4138 . . 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 3512 . . . 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 2443 . . . . . . . . 9  |-  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  =  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
43rnmptss 5887 . . . . . . . 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 11107 . . . . . . . . . . 11  |- +oo  e.  RR*
6 icossre 11391 . . . . . . . . . . 11  |-  ( ( e  e.  RR  /\ +oo  e.  RR* )  ->  (
e [,) +oo )  C_  RR )
75, 6mpan2 671 . . . . . . . . . 10  |-  ( e  e.  RR  ->  (
e [,) +oo )  C_  RR )
8 xpss1 4963 . . . . . . . . . 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 6131 . . . . . . . . . . 11  |-  ( e [,) +oo )  e. 
_V
11 reex 9388 . . . . . . . . . . 11  |-  RR  e.  _V
1210, 11xpex 6523 . . . . . . . . . 10  |-  ( ( e [,) +oo )  X.  RR )  e.  _V
1312elpw 3881 . . . . . . . . 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 2800 . . . . . . 7  |-  ran  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) 
C_  ~P ( RR  X.  RR )
1615sseli 3367 . . . . . 6  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
1716elpwid 3885 . . . . 5  |-  ( z  e.  ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )  -> 
z  C_  ( RR  X.  RR ) )
18 eqid 2443 . . . . . . . . 9  |-  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  =  ( f  e.  RR  |->  ( RR  X.  (
f [,) +oo )
) )
1918rnmptss 5887 . . . . . . . 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 11391 . . . . . . . . . . 11  |-  ( ( f  e.  RR  /\ +oo  e.  RR* )  ->  (
f [,) +oo )  C_  RR )
215, 20mpan2 671 . . . . . . . . . 10  |-  ( f  e.  RR  ->  (
f [,) +oo )  C_  RR )
22 xpss2 4964 . . . . . . . . . 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 6131 . . . . . . . . . . 11  |-  ( f [,) +oo )  e. 
_V
2511, 24xpex 6523 . . . . . . . . . 10  |-  ( RR 
X.  ( f [,) +oo ) )  e.  _V
2625elpw 3881 . . . . . . . . 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 2800 . . . . . . 7  |-  ran  (
f  e.  RR  |->  ( RR  X.  ( f [,) +oo ) ) )  C_  ~P ( RR  X.  RR )
2928sseli 3367 . . . . . 6  |-  ( z  e.  ran  ( f  e.  RR  |->  ( RR 
X.  ( f [,) +oo ) ) )  -> 
z  e.  ~P ( RR  X.  RR ) )
3029elpwid 3885 . . . . 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 2801 . 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 5469 . . . . . 6  |-  Fun  (
e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
35 rexr 9444 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  e. 
RR* )
365a1i 11 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  -> +oo  e.  RR* )
37 ltpnf 11117 . . . . . . . . . . 11  |-  ( ( 1st `  z )  e.  RR  ->  ( 1st `  z )  < +oo )
38 lbico1 11365 . . . . . . . . . . 11  |-  ( ( ( 1st `  z
)  e.  RR*  /\ +oo  e.  RR*  /\  ( 1st `  z )  < +oo )  ->  ( 1st `  z
)  e.  ( ( 1st `  z ) [,) +oo ) )
3935, 36, 37, 38syl3anc 1218 . . . . . . . . . 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 6624 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  <->  ( z  e.  ( _V  X.  _V )  /\  ( ( 1st `  z )  e.  RR  /\  ( 2nd `  z
)  e.  RR ) ) )
43 elxp7 6624 . . . . . . . 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 6621 . . . . . . . 8  |-  ( z  e.  ( RR  X.  RR )  ->  ( 1st `  z )  e.  RR )
46 oveq1 6113 . . . . . . . . . 10  |-  ( e  =  ( 1st `  z
)  ->  ( e [,) +oo )  =  ( ( 1st `  z
) [,) +oo )
)
4746xpeq1d 4878 . . . . . . . . 9  |-  ( e  =  ( 1st `  z
)  ->  ( (
e [,) +oo )  X.  RR )  =  ( ( ( 1st `  z
) [,) +oo )  X.  RR ) )
48 ovex 6131 . . . . . . . . . 10  |-  ( ( 1st `  z ) [,) +oo )  e. 
_V
4948, 11xpex 6523 . . . . . . . . 9  |-  ( ( ( 1st `  z
) [,) +oo )  X.  RR )  e.  _V
5047, 3, 49fvmpt 5789 . . . . . . . 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 2520 . . . . . 6  |-  ( z  e.  ( RR  X.  RR )  ->  z  e.  ( ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) ) `  ( 1st `  z ) ) )
53 elunirn2 25981 . . . . . 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 3375 . . . 4  |-  ( RR 
X.  RR )  C_  U.
ran  ( e  e.  RR  |->  ( ( e [,) +oo )  X.  RR ) )
56 ssun3 3536 . . . 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 4125 . . 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 3404 . 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 3387 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 1369    e. wcel 1756   _Vcvv 2987    u. cun 3341    C_ wss 3343   ~Pcpw 3875   U.cuni 4106   class class class wbr 4307    e. cmpt 4365    X. cxp 4853   ran crn 4856   Fun wfun 5427   ` cfv 5433  (class class class)co 6106   1stc1st 6590   2ndc2nd 6591   RRcr 9296   +oocpnf 9430   RR*cxr 9432    < clt 9433   [,)cico 11317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-pre-lttri 9371  ax-pre-lttrn 9372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-po 4656  df-so 4657  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-1st 6592  df-2nd 6593  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-ico 11321
This theorem is referenced by:  sxbrsigalem3  26702  sxbrsigalem2  26716
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