Theorem sxbrsigalem3 30334

Description: The sigma-algebra generated by the closed half-spaces of ( RR X. RR ) is a subset of the sigma-algebra generated by the closed sets of ( RR X. RR ). (Contributed by Thierry Arnoux, 11-Oct-2017.)

Hypothesis

Ref	Expression
sxbrsiga.0	|- J = ( topGen ` ran (,) )

Assertion

Ref	Expression
sxbrsigalem3	|- (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) C_ (sigaGen ` ( Clsd ` ( J tX J ) ) )

Distinct variable group:   e, f

Allowed substitution hints:   J( e, f)

Proof of Theorem sxbrsigalem3

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sxbrsigalem0 30333	. . 3 |- U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) = ( RR X. RR )
2		sxbrsiga.0	. . . . . 6 |- J = ( topGen ` ran (,) )
3		retop 22565	. . . . . 6 |- ( topGen ` ran (,) ) e. Top
4	2, 3	eqeltri 2697	. . . . 5 |- J e. Top
5	4, 4	txtopi 21393	. . . 4 |- ( J tX J ) e. Top
6		uniretop 22566	. . . . . 6 |- RR = U. ( topGen ` ran (,) )
7	2	unieqi 4445	. . . . . 6 |- U. J = U. ( topGen ` ran (,) )
8	6, 7	eqtr4i 2647	. . . . 5 |- RR = U. J
9	4, 4, 8, 8	txunii 21396	. . . 4 |- ( RR X. RR ) = U. ( J tX J )
10	5, 9	unicls 29949	. . 3 |- U. ( Clsd ` ( J tX J ) ) = ( RR X. RR )
11	1, 10	eqtr4i 2647	. 2 |- U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) = U. ( Clsd ` ( J tX J ) )
12		ovex 6678	. . . . . . 7 |- ( e [,) +oo ) e. _V
13		reex 10027	. . . . . . 7 |- RR e. _V
14	12, 13	xpex 6962	. . . . . 6 |- ( ( e [,) +oo ) X. RR ) e. _V
15		eqid 2622	. . . . . 6 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) = ( e e. RR |-> ( ( e [,) +oo ) X. RR ) )
16	14, 15	fnmpti 6022	. . . . 5 |- ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) Fn RR
17		oveq1 6657	. . . . . . . . 9 |- ( e = u -> ( e [,) +oo ) = ( u [,) +oo ) )
18	17	xpeq1d 5138	. . . . . . . 8 |- ( e = u -> ( ( e [,) +oo ) X. RR ) = ( ( u [,) +oo ) X. RR ) )
19		ovex 6678	. . . . . . . . 9 |- ( u [,) +oo ) e. _V
20	19, 13	xpex 6962	. . . . . . . 8 |- ( ( u [,) +oo ) X. RR ) e. _V
21	18, 15, 20	fvmpt 6282	. . . . . . 7 |- ( u e. RR -> ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) ` u ) = ( ( u [,) +oo ) X. RR ) )
22		icopnfcld 22571	. . . . . . . . 9 |- ( u e. RR -> ( u [,) +oo ) e. ( Clsd ` ( topGen ` ran (,) ) ) )
23	2	fveq2i 6194	. . . . . . . . 9 |- ( Clsd ` J ) = ( Clsd ` ( topGen ` ran (,) ) )
24	22, 23	syl6eleqr 2712	. . . . . . . 8 |- ( u e. RR -> ( u [,) +oo ) e. ( Clsd ` J ) )
25		dif0 3950	. . . . . . . . 9 |- ( RR \ (/) ) = RR
26		0opn 20709	. . . . . . . . . . 11 |- ( J e. Top -> (/) e. J )
27	4, 26	ax-mp 5	. . . . . . . . . 10 |- (/) e. J
28	8	opncld 20837	. . . . . . . . . 10 |- ( ( J e. Top /\ (/) e. J ) -> ( RR \ (/) ) e. ( Clsd ` J ) )
29	4, 27, 28	mp2an 708	. . . . . . . . 9 |- ( RR \ (/) ) e. ( Clsd ` J )
30	25, 29	eqeltrri 2698	. . . . . . . 8 |- RR e. ( Clsd ` J )
31		txcld 21406	. . . . . . . 8 |- ( ( ( u [,) +oo ) e. ( Clsd ` J ) /\ RR e. ( Clsd ` J ) ) -> ( ( u [,) +oo ) X. RR ) e. ( Clsd ` ( J tX J ) ) )
32	24, 30, 31	sylancl 694	. . . . . . 7 |- ( u e. RR -> ( ( u [,) +oo ) X. RR ) e. ( Clsd ` ( J tX J ) ) )
33	21, 32	eqeltrd 2701	. . . . . 6 |- ( u e. RR -> ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) ` u ) e. ( Clsd ` ( J tX J ) ) )
34	33	rgen 2922	. . . . 5 |- A. u e. RR ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) ` u ) e. ( Clsd ` ( J tX J ) )
35		fnfvrnss 6390	. . . . 5 |- ( ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) Fn RR /\ A. u e. RR ( ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) ` u ) e. ( Clsd ` ( J tX J ) ) ) -> ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) C_ ( Clsd ` ( J tX J ) ) )
36	16, 34, 35	mp2an 708	. . . 4 |- ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) C_ ( Clsd ` ( J tX J ) )
37		ovex 6678	. . . . . . 7 |- ( f [,) +oo ) e. _V
38	13, 37	xpex 6962	. . . . . 6 |- ( RR X. ( f [,) +oo ) ) e. _V
39		eqid 2622	. . . . . 6 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) = ( f e. RR |-> ( RR X. ( f [,) +oo ) ) )
40	38, 39	fnmpti 6022	. . . . 5 |- ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) Fn RR
41		oveq1 6657	. . . . . . . . 9 |- ( f = v -> ( f [,) +oo ) = ( v [,) +oo ) )
42	41	xpeq2d 5139	. . . . . . . 8 |- ( f = v -> ( RR X. ( f [,) +oo ) ) = ( RR X. ( v [,) +oo ) ) )
43		ovex 6678	. . . . . . . . 9 |- ( v [,) +oo ) e. _V
44	13, 43	xpex 6962	. . . . . . . 8 |- ( RR X. ( v [,) +oo ) ) e. _V
45	42, 39, 44	fvmpt 6282	. . . . . . 7 |- ( v e. RR -> ( ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ` v ) = ( RR X. ( v [,) +oo ) ) )
46		icopnfcld 22571	. . . . . . . . 9 |- ( v e. RR -> ( v [,) +oo ) e. ( Clsd ` ( topGen ` ran (,) ) ) )
47	46, 23	syl6eleqr 2712	. . . . . . . 8 |- ( v e. RR -> ( v [,) +oo ) e. ( Clsd ` J ) )
48		txcld 21406	. . . . . . . 8 |- ( ( RR e. ( Clsd ` J ) /\ ( v [,) +oo ) e. ( Clsd ` J ) ) -> ( RR X. ( v [,) +oo ) ) e. ( Clsd ` ( J tX J ) ) )
49	30, 47, 48	sylancr 695	. . . . . . 7 |- ( v e. RR -> ( RR X. ( v [,) +oo ) ) e. ( Clsd ` ( J tX J ) ) )
50	45, 49	eqeltrd 2701	. . . . . 6 |- ( v e. RR -> ( ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ` v ) e. ( Clsd ` ( J tX J ) ) )
51	50	rgen 2922	. . . . 5 |- A. v e. RR ( ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ` v ) e. ( Clsd ` ( J tX J ) )
52		fnfvrnss 6390	. . . . 5 |- ( ( ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) Fn RR /\ A. v e. RR ( ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ` v ) e. ( Clsd ` ( J tX J ) ) ) -> ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) C_ ( Clsd ` ( J tX J ) ) )
53	40, 51, 52	mp2an 708	. . . 4 |- ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) C_ ( Clsd ` ( J tX J ) )
54	36, 53	unssi 3788	. . 3 |- ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) C_ ( Clsd ` ( J tX J ) )
55		fvex 6201	. . . 4 |- ( Clsd ` ( J tX J ) ) e. _V
56		sssigagen 30208	. . . 4 |- ( ( Clsd ` ( J tX J ) ) e. _V -> ( Clsd ` ( J tX J ) ) C_ (sigaGen ` ( Clsd ` ( J tX J ) ) ) )
57	55, 56	ax-mp 5	. . 3 |- ( Clsd ` ( J tX J ) ) C_ (sigaGen ` ( Clsd ` ( J tX J ) ) )
58	54, 57	sstri 3612	. 2 |- ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) C_ (sigaGen ` ( Clsd ` ( J tX J ) ) )
59		sigagenss2 30213	. 2 |- ( ( U. ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) = U. ( Clsd ` ( J tX J ) ) /\ ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) C_ (sigaGen ` ( Clsd ` ( J tX J ) ) ) /\ ( Clsd ` ( J tX J ) ) e. _V ) -> (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) C_ (sigaGen ` ( Clsd ` ( J tX J ) ) ) )
60	11, 58, 55, 59	mp3an 1424	1 |- (sigaGen ` ( ran ( e e. RR |-> ( ( e [,) +oo ) X. RR ) ) u. ran ( f e. RR |-> ( RR X. ( f [,) +oo ) ) ) ) ) C_ (sigaGen ` ( Clsd ` ( J tX J ) ) )

Syntax hints:   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   ran crn 5115   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   RRcr 9935   +oocpnf 10071   (,)cioo 12175   [,)cico 12177   topGenctg 16098   Topctop 20698   Clsdccld 20820   tX ctx 21363  sigaGencsigagen 30201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-ioo 12179  df-ico 12181  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-tx 21365  df-siga 30171  df-sigagen 30202

This theorem is referenced by:  sxbrsigalem4  30349