Theorem issmfdmpt 40957

Description: A sufficient condition for " F being a measurable function w.r.t. to the sigma-algebra S". (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
issmfdmpt.x	|- F/ x ph
issmfdmpt.a	|- F/ a ph
issmfdmpt.s	|- ( ph -> S e. SAlg )
issmfdmpt.i	|- ( ph -> A C_ U. S )
issmfdmpt.b	|- ( ( ph /\ x e. A ) -> B e. RR )
issmfdmpt.p	|- ( ( ph /\ a e. RR ) -> { x e. A | B < a } e. ( S ↾t A ) )

Assertion

Ref	Expression
issmfdmpt	|- ( ph -> ( x e. A |-> B ) e. (SMblFn ` S ) )

Distinct variable groups:   A, a, x   B, a   S, a

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

Proof of Theorem issmfdmpt

Step	Hyp	Ref	Expression
1		nfmpt1 4747	. 2 |- F/_ x ( x e. A |-> B )
2		issmfdmpt.a	. 2 |- F/ a ph
3		issmfdmpt.s	. 2 |- ( ph -> S e. SAlg )
4		issmfdmpt.i	. 2 |- ( ph -> A C_ U. S )
5		issmfdmpt.x	. . 3 |- F/ x ph
6		issmfdmpt.b	. . 3 |- ( ( ph /\ x e. A ) -> B e. RR )
7		eqid 2622	. . 3 |- ( x e. A |-> B ) = ( x e. A |-> B )
8	5, 6, 7	fmptdf 6387	. 2 |- ( ph -> ( x e. A |-> B ) : A --> RR )
9		eqidd 2623	. . . . . . . . 9 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
10	9, 6	fvmpt2d 6293	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
11	10	breq1d 4663	. . . . . . 7 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) < a <-> B < a ) )
12	11	ex 450	. . . . . 6 |- ( ph -> ( x e. A -> ( ( ( x e. A |-> B ) ` x ) < a <-> B < a ) ) )
13	5, 12	ralrimi 2957	. . . . 5 |- ( ph -> A. x e. A ( ( ( x e. A |-> B ) ` x ) < a <-> B < a ) )
14		rabbi 3120	. . . . 5 |- ( A. x e. A ( ( ( x e. A |-> B ) ` x ) < a <-> B < a ) <-> { x e. A | ( ( x e. A |-> B ) ` x ) < a } = { x e. A | B < a } )
15	13, 14	sylib 208	. . . 4 |- ( ph -> { x e. A | ( ( x e. A |-> B ) ` x ) < a } = { x e. A | B < a } )
16	15	adantr 481	. . 3 |- ( ( ph /\ a e. RR ) -> { x e. A | ( ( x e. A |-> B ) ` x ) < a } = { x e. A | B < a } )
17		issmfdmpt.p	. . 3 |- ( ( ph /\ a e. RR ) -> { x e. A | B < a } e. ( S ↾t A ) )
18	16, 17	eqeltrd 2701	. 2 |- ( ( ph /\ a e. RR ) -> { x e. A | ( ( x e. A |-> B ) ` x ) < a } e. ( S ↾t A ) )
19	1, 2, 3, 4, 8, 18	issmfdf 40946	1 |- ( ph -> ( x e. A |-> B ) e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   ` 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:  smfadd  40973  smfrec  40996  smfmul  41002