Theorem smfpimltxr 40956

Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smfpimltxr.x	|- F/_ x F
smfpimltxr.s	|- ( ph -> S e. SAlg )
smfpimltxr.f	|- ( ph -> F e. (SMblFn ` S ) )
smfpimltxr.d	|- D = dom F
smfpimltxr.a	|- ( ph -> A e. RR* )

Assertion

Ref	Expression
smfpimltxr	|- ( ph -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )

Distinct variable groups:   x, A   x, D

Allowed substitution hints:   ph( x)   S( x)   F( x)

Proof of Theorem smfpimltxr

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 4657	. . . . . 6 |- ( A = +oo -> ( ( F ` x ) < A <-> ( F ` x ) < +oo ) )
2	1	rabbidv 3189	. . . . 5 |- ( A = +oo -> { x e. D | ( F ` x ) < A } = { x e. D | ( F ` x ) < +oo } )
3	2	adantl 482	. . . 4 |- ( ( ph /\ A = +oo ) -> { x e. D | ( F ` x ) < A } = { x e. D | ( F ` x ) < +oo } )
4		smfpimltxr.x	. . . . . 6 |- F/_ x F
5		smfpimltxr.f	. . . . . . . 8 |- ( ph -> F e. (SMblFn ` S ) )
6		smfpimltxr.s	. . . . . . . . 9 |- ( ph -> S e. SAlg )
7		smfpimltxr.d	. . . . . . . . 9 |- D = dom F
8	4, 6, 7	issmff 40943	. . . . . . . 8 |- ( ph -> ( F e. (SMblFn ` S ) <-> ( D C_ U. S /\ F : D --> RR /\ A. a e. RR { x e. D | ( F ` x ) < a } e. ( S ↾t D ) ) ) )
9	5, 8	mpbid 222	. . . . . . 7 |- ( ph -> ( D C_ U. S /\ F : D --> RR /\ A. a e. RR { x e. D | ( F ` x ) < a } e. ( S ↾t D ) ) )
10	9	simp2d 1074	. . . . . 6 |- ( ph -> F : D --> RR )
11	4, 10	pimltpnf2 40923	. . . . 5 |- ( ph -> { x e. D | ( F ` x ) < +oo } = D )
12	11	adantr 481	. . . 4 |- ( ( ph /\ A = +oo ) -> { x e. D | ( F ` x ) < +oo } = D )
13		eqidd 2623	. . . 4 |- ( ( ph /\ A = +oo ) -> D = D )
14	3, 12, 13	3eqtrd 2660	. . 3 |- ( ( ph /\ A = +oo ) -> { x e. D | ( F ` x ) < A } = D )
15	9	simp1d 1073	. . . . . . 7 |- ( ph -> D C_ U. S )
16	6, 15	restuni4 39304	. . . . . 6 |- ( ph -> U. ( S ↾t D ) = D )
17	16	eqcomd 2628	. . . . 5 |- ( ph -> D = U. ( S ↾t D ) )
18	5	dmexd 39422	. . . . . . . 8 |- ( ph -> dom F e. _V )
19	7, 18	syl5eqel 2705	. . . . . . 7 |- ( ph -> D e. _V )
20		eqid 2622	. . . . . . 7 |- ( S ↾t D ) = ( S ↾t D )
21	6, 19, 20	subsalsal 40577	. . . . . 6 |- ( ph -> ( S ↾t D ) e. SAlg )
22	21	salunid 40571	. . . . 5 |- ( ph -> U. ( S ↾t D ) e. ( S ↾t D ) )
23	17, 22	eqeltrd 2701	. . . 4 |- ( ph -> D e. ( S ↾t D ) )
24	23	adantr 481	. . 3 |- ( ( ph /\ A = +oo ) -> D e. ( S ↾t D ) )
25	14, 24	eqeltrd 2701	. 2 |- ( ( ph /\ A = +oo ) -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )
26		neqne 2802	. . . 4 |- ( -. A = +oo -> A =/= +oo )
27	26	adantl 482	. . 3 |- ( ( ph /\ -. A = +oo ) -> A =/= +oo )
28		breq2 4657	. . . . . . . . 9 |- ( A = -oo -> ( ( F ` x ) < A <-> ( F ` x ) < -oo ) )
29	28	rabbidv 3189	. . . . . . . 8 |- ( A = -oo -> { x e. D | ( F ` x ) < A } = { x e. D | ( F ` x ) < -oo } )
30	29	adantl 482	. . . . . . 7 |- ( ( ph /\ A = -oo ) -> { x e. D | ( F ` x ) < A } = { x e. D | ( F ` x ) < -oo } )
31	10	adantr 481	. . . . . . . 8 |- ( ( ph /\ A = -oo ) -> F : D --> RR )
32	4, 31	pimltmnf2 40911	. . . . . . 7 |- ( ( ph /\ A = -oo ) -> { x e. D | ( F ` x ) < -oo } = (/) )
33	30, 32	eqtrd 2656	. . . . . 6 |- ( ( ph /\ A = -oo ) -> { x e. D | ( F ` x ) < A } = (/) )
34	21	0sald 40568	. . . . . . 7 |- ( ph -> (/) e. ( S ↾t D ) )
35	34	adantr 481	. . . . . 6 |- ( ( ph /\ A = -oo ) -> (/) e. ( S ↾t D ) )
36	33, 35	eqeltrd 2701	. . . . 5 |- ( ( ph /\ A = -oo ) -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )
37	36	adantlr 751	. . . 4 |- ( ( ( ph /\ A =/= +oo ) /\ A = -oo ) -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )
38		simpll 790	. . . . 5 |- ( ( ( ph /\ A =/= +oo ) /\ -. A = -oo ) -> ph )
39		smfpimltxr.a	. . . . . . 7 |- ( ph -> A e. RR* )
40	38, 39	syl 17	. . . . . 6 |- ( ( ( ph /\ A =/= +oo ) /\ -. A = -oo ) -> A e. RR* )
41		neqne 2802	. . . . . . 7 |- ( -. A = -oo -> A =/= -oo )
42	41	adantl 482	. . . . . 6 |- ( ( ( ph /\ A =/= +oo ) /\ -. A = -oo ) -> A =/= -oo )
43		simplr 792	. . . . . 6 |- ( ( ( ph /\ A =/= +oo ) /\ -. A = -oo ) -> A =/= +oo )
44	40, 42, 43	xrred 39581	. . . . 5 |- ( ( ( ph /\ A =/= +oo ) /\ -. A = -oo ) -> A e. RR )
45	6	adantr 481	. . . . . 6 |- ( ( ph /\ A e. RR ) -> S e. SAlg )
46	5	adantr 481	. . . . . 6 |- ( ( ph /\ A e. RR ) -> F e. (SMblFn ` S ) )
47		simpr 477	. . . . . 6 |- ( ( ph /\ A e. RR ) -> A e. RR )
48	4, 45, 46, 7, 47	smfpreimaltf 40945	. . . . 5 |- ( ( ph /\ A e. RR ) -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )
49	38, 44, 48	syl2anc 693	. . . 4 |- ( ( ( ph /\ A =/= +oo ) /\ -. A = -oo ) -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )
50	37, 49	pm2.61dan 832	. . 3 |- ( ( ph /\ A =/= +oo ) -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )
51	27, 50	syldan 487	. 2 |- ( ( ph /\ -. A = +oo ) -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )
52	25, 51	pm2.61dan 832	1 |- ( ph -> { x e. D | ( F ` x ) < A } e. ( S ↾t D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U.cuni 4436   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   ↾_t crest 16081  SAlgcsalg 40528  SMblFncsmblfn 40909

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-ac2 9285  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-se 5074  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-isom 5897  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-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-ioo 12179  df-ico 12181  df-rest 16083  df-salg 40529  df-smblfn 40910

This theorem is referenced by:  smfpimltxrmpt  40967