Theorem smfpimgtxr 40988

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

Hypotheses

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

Assertion

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

Distinct variable groups:   x, A   x, D

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

Proof of Theorem smfpimgtxr

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. . . . . 6 |- ( A = -oo -> ( A < ( F ` x ) <-> -oo < ( F ` x ) ) )
2	1	rabbidv 3189	. . . . 5 |- ( A = -oo -> { x e. D | A < ( F ` x ) } = { x e. D | -oo < ( F ` x ) } )
3	2	adantl 482	. . . 4 |- ( ( ph /\ A = -oo ) -> { x e. D | A < ( F ` x ) } = { x e. D | -oo < ( F ` x ) } )
4		smfpimgtxr.d	. . . . . . . . 9 |- D = dom F
5		smfpimgtxr.x	. . . . . . . . . 10 |- F/_ x F
6	5	nfdm 5367	. . . . . . . . 9 |- F/_ x dom F
7	4, 6	nfcxfr 2762	. . . . . . . 8 |- F/_ x D
8		nfcv 2764	. . . . . . . 8 |- F/_ y D
9		nfv 1843	. . . . . . . 8 |- F/ y -oo < ( F ` x )
10		nfcv 2764	. . . . . . . . 9 |- F/_ x -oo
11		nfcv 2764	. . . . . . . . 9 |- F/_ x <
12		nfcv 2764	. . . . . . . . . 10 |- F/_ x y
13	5, 12	nffv 6198	. . . . . . . . 9 |- F/_ x ( F ` y )
14	10, 11, 13	nfbr 4699	. . . . . . . 8 |- F/ x -oo < ( F ` y )
15		fveq2 6191	. . . . . . . . 9 |- ( x = y -> ( F ` x ) = ( F ` y ) )
16	15	breq2d 4665	. . . . . . . 8 |- ( x = y -> ( -oo < ( F ` x ) <-> -oo < ( F ` y ) ) )
17	7, 8, 9, 14, 16	cbvrab 3198	. . . . . . 7 |- { x e. D | -oo < ( F ` x ) } = { y e. D | -oo < ( F ` y ) }
18	17	a1i 11	. . . . . 6 |- ( ph -> { x e. D | -oo < ( F ` x ) } = { y e. D | -oo < ( F ` y ) } )
19		nfv 1843	. . . . . . 7 |- F/ y ph
20		smfpimgtxr.s	. . . . . . . . . 10 |- ( ph -> S e. SAlg )
21		smfpimgtxr.f	. . . . . . . . . 10 |- ( ph -> F e. (SMblFn ` S ) )
22	20, 21, 4	smff 40941	. . . . . . . . 9 |- ( ph -> F : D --> RR )
23	22	adantr 481	. . . . . . . 8 |- ( ( ph /\ y e. D ) -> F : D --> RR )
24		simpr 477	. . . . . . . 8 |- ( ( ph /\ y e. D ) -> y e. D )
25	23, 24	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ y e. D ) -> ( F ` y ) e. RR )
26	19, 25	pimgtmnf 40932	. . . . . 6 |- ( ph -> { y e. D | -oo < ( F ` y ) } = D )
27		eqidd 2623	. . . . . 6 |- ( ph -> D = D )
28	18, 26, 27	3eqtrd 2660	. . . . 5 |- ( ph -> { x e. D | -oo < ( F ` x ) } = D )
29	28	adantr 481	. . . 4 |- ( ( ph /\ A = -oo ) -> { x e. D | -oo < ( F ` x ) } = D )
30	3, 29	eqtrd 2656	. . 3 |- ( ( ph /\ A = -oo ) -> { x e. D | A < ( F ` x ) } = D )
31	20, 21, 4	smfdmss 40942	. . . . . . 7 |- ( ph -> D C_ U. S )
32	20, 31	restuni4 39304	. . . . . 6 |- ( ph -> U. ( S ↾t D ) = D )
33	32	eqcomd 2628	. . . . 5 |- ( ph -> D = U. ( S ↾t D ) )
34	21	dmexd 39422	. . . . . . . 8 |- ( ph -> dom F e. _V )
35	4, 34	syl5eqel 2705	. . . . . . 7 |- ( ph -> D e. _V )
36		eqid 2622	. . . . . . 7 |- ( S ↾t D ) = ( S ↾t D )
37	20, 35, 36	subsalsal 40577	. . . . . 6 |- ( ph -> ( S ↾t D ) e. SAlg )
38	37	salunid 40571	. . . . 5 |- ( ph -> U. ( S ↾t D ) e. ( S ↾t D ) )
39	33, 38	eqeltrd 2701	. . . 4 |- ( ph -> D e. ( S ↾t D ) )
40	39	adantr 481	. . 3 |- ( ( ph /\ A = -oo ) -> D e. ( S ↾t D ) )
41	30, 40	eqeltrd 2701	. 2 |- ( ( ph /\ A = -oo ) -> { x e. D | A < ( F ` x ) } e. ( S ↾t D ) )
42		neqne 2802	. . . 4 |- ( -. A = -oo -> A =/= -oo )
43	42	adantl 482	. . 3 |- ( ( ph /\ -. A = -oo ) -> A =/= -oo )
44		breq1 4656	. . . . . . . . 9 |- ( A = +oo -> ( A < ( F ` x ) <-> +oo < ( F ` x ) ) )
45	44	rabbidv 3189	. . . . . . . 8 |- ( A = +oo -> { x e. D | A < ( F ` x ) } = { x e. D | +oo < ( F ` x ) } )
46	45	adantl 482	. . . . . . 7 |- ( ( ph /\ A = +oo ) -> { x e. D | A < ( F ` x ) } = { x e. D | +oo < ( F ` x ) } )
47	5, 22	pimgtpnf2 40917	. . . . . . . 8 |- ( ph -> { x e. D | +oo < ( F ` x ) } = (/) )
48	47	adantr 481	. . . . . . 7 |- ( ( ph /\ A = +oo ) -> { x e. D | +oo < ( F ` x ) } = (/) )
49	46, 48	eqtrd 2656	. . . . . 6 |- ( ( ph /\ A = +oo ) -> { x e. D | A < ( F ` x ) } = (/) )
50	37	0sald 40568	. . . . . . 7 |- ( ph -> (/) e. ( S ↾t D ) )
51	50	adantr 481	. . . . . 6 |- ( ( ph /\ A = +oo ) -> (/) e. ( S ↾t D ) )
52	49, 51	eqeltrd 2701	. . . . 5 |- ( ( ph /\ A = +oo ) -> { x e. D | A < ( F ` x ) } e. ( S ↾t D ) )
53	52	adantlr 751	. . . 4 |- ( ( ( ph /\ A =/= -oo ) /\ A = +oo ) -> { x e. D | A < ( F ` x ) } e. ( S ↾t D ) )
54		simpll 790	. . . . 5 |- ( ( ( ph /\ A =/= -oo ) /\ -. A = +oo ) -> ph )
55		smfpimgtxr.a	. . . . . . 7 |- ( ph -> A e. RR* )
56	54, 55	syl 17	. . . . . 6 |- ( ( ( ph /\ A =/= -oo ) /\ -. A = +oo ) -> A e. RR* )
57		simplr 792	. . . . . 6 |- ( ( ( ph /\ A =/= -oo ) /\ -. A = +oo ) -> A =/= -oo )
58		neqne 2802	. . . . . . 7 |- ( -. A = +oo -> A =/= +oo )
59	58	adantl 482	. . . . . 6 |- ( ( ( ph /\ A =/= -oo ) /\ -. A = +oo ) -> A =/= +oo )
60	56, 57, 59	xrred 39581	. . . . 5 |- ( ( ( ph /\ A =/= -oo ) /\ -. A = +oo ) -> A e. RR )
61	20	adantr 481	. . . . . 6 |- ( ( ph /\ A e. RR ) -> S e. SAlg )
62	21	adantr 481	. . . . . 6 |- ( ( ph /\ A e. RR ) -> F e. (SMblFn ` S ) )
63		simpr 477	. . . . . 6 |- ( ( ph /\ A e. RR ) -> A e. RR )
64	5, 61, 62, 4, 63	smfpreimagtf 40976	. . . . 5 |- ( ( ph /\ A e. RR ) -> { x e. D | A < ( F ` x ) } e. ( S ↾t D ) )
65	54, 60, 64	syl2anc 693	. . . 4 |- ( ( ( ph /\ A =/= -oo ) /\ -. A = +oo ) -> { x e. D | A < ( F ` x ) } e. ( S ↾t D ) )
66	53, 65	pm2.61dan 832	. . 3 |- ( ( ph /\ A =/= -oo ) -> { x e. D | A < ( F ` x ) } e. ( S ↾t D ) )
67	43, 66	syldan 487	. 2 |- ( ( ph /\ -. A = -oo ) -> { x e. D | A < ( F ` x ) } e. ( S ↾t D ) )
68	41, 67	pm2.61dan 832	1 |- ( ph -> { x e. D | A < ( F ` x ) } e. ( S ↾t D ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   {crab 2916   _Vcvv 3200   (/)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  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-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-iin 4523  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-sup 8348  df-inf 8349  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-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-fl 12593  df-rest 16083  df-salg 40529  df-smblfn 40910

This theorem is referenced by:  smfpimgtxrmpt  40992