Theorem issmf 40937

Description: The predicate " F is a real-valued measurable function w.r.t. to the sigma-algebra S". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of F is required to be a subset of the underlying set of S. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
issmf.s	|- ( ph -> S e. SAlg )
issmf.d	|- D = dom F

Assertion

Ref	Expression
issmf	|- ( 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 ) ) ) )

Distinct variable groups:   D, a, x   F, a, x   S, a

Allowed substitution hints:   ph( x, a)   S( x)

Proof of Theorem issmf

Dummy variables b y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		issmf.s	. . 3 |- ( ph -> S e. SAlg )
2		issmf.d	. . 3 |- D = dom F
3	1, 2	issmflem 40936	. 2 |- ( ph -> ( F e. (SMblFn ` S ) <-> ( D C_ U. S /\ F : D --> RR /\ A. b e. RR { y e. D | ( F ` y ) < b } e. ( S ↾t D ) ) ) )
4		breq2 4657	. . . . . . . 8 |- ( b = a -> ( ( F ` y ) < b <-> ( F ` y ) < a ) )
5	4	rabbidv 3189	. . . . . . 7 |- ( b = a -> { y e. D | ( F ` y ) < b } = { y e. D | ( F ` y ) < a } )
6	5	eleq1d 2686	. . . . . 6 |- ( b = a -> ( { y e. D | ( F ` y ) < b } e. ( S ↾t D ) <-> { y e. D | ( F ` y ) < a } e. ( S ↾t D ) ) )
7		fveq2 6191	. . . . . . . . . 10 |- ( y = x -> ( F ` y ) = ( F ` x ) )
8	7	breq1d 4663	. . . . . . . . 9 |- ( y = x -> ( ( F ` y ) < a <-> ( F ` x ) < a ) )
9	8	cbvrabv 3199	. . . . . . . 8 |- { y e. D | ( F ` y ) < a } = { x e. D | ( F ` x ) < a }
10	9	eleq1i 2692	. . . . . . 7 |- ( { y e. D | ( F ` y ) < a } e. ( S ↾t D ) <-> { x e. D | ( F ` x ) < a } e. ( S ↾t D ) )
11	10	a1i 11	. . . . . 6 |- ( b = a -> ( { y e. D | ( F ` y ) < a } e. ( S ↾t D ) <-> { x e. D | ( F ` x ) < a } e. ( S ↾t D ) ) )
12	6, 11	bitrd 268	. . . . 5 |- ( b = a -> ( { y e. D | ( F ` y ) < b } e. ( S ↾t D ) <-> { x e. D | ( F ` x ) < a } e. ( S ↾t D ) ) )
13	12	cbvralv 3171	. . . 4 |- ( A. b e. RR { y e. D | ( F ` y ) < b } e. ( S ↾t D ) <-> A. a e. RR { x e. D | ( F ` x ) < a } e. ( S ↾t D ) )
14	13	3anbi3i 1255	. . 3 |- ( ( D C_ U. S /\ F : D --> RR /\ A. b e. RR { y e. D | ( F ` y ) < b } e. ( S ↾t D ) ) <-> ( D C_ U. S /\ F : D --> RR /\ A. a e. RR { x e. D | ( F ` x ) < a } e. ( S ↾t D ) ) )
15	14	a1i 11	. 2 |- ( ph -> ( ( D C_ U. S /\ F : D --> RR /\ A. b e. RR { y e. D | ( F ` y ) < b } e. ( S ↾t D ) ) <-> ( D C_ U. S /\ F : D --> RR /\ A. a e. RR { x e. D | ( F ` x ) < a } e. ( S ↾t D ) ) ) )
16	3, 15	bitrd 268	1 |- ( 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 ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   U.cuni 4436   class class class wbr 4653   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   < 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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioo 12179  df-ico 12181  df-smblfn 40910

This theorem is referenced by:  smfpreimalt  40940  smff  40941  smfdmss  40942  issmff  40943  issmfd  40944  issmflelem  40953  issmfgtlem  40964  issmfgelem  40977