Theorem smfpimltmpt 40955

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
smfpimltmpt.x	|- F/ x ph
smfpimltmpt.s	|- ( ph -> S e. SAlg )
smfpimltmpt.b	|- ( ( ph /\ x e. A ) -> B e. V )
smfpimltmpt.f	|- ( ph -> ( x e. A |-> B ) e. (SMblFn ` S ) )
smfpimltmpt.r	|- ( ph -> R e. RR )

Assertion

Ref	Expression
smfpimltmpt	|- ( ph -> { x e. A | B < R } e. ( S ↾t A ) )

Distinct variable groups:   x, A   x, R

Allowed substitution hints:   ph( x)   B( x)   S( x)   V( x)

Proof of Theorem smfpimltmpt

Step	Hyp	Ref	Expression
1		nfmpt1 4747	. . 3 |- F/_ x ( x e. A |-> B )
2		smfpimltmpt.s	. . 3 |- ( ph -> S e. SAlg )
3		smfpimltmpt.f	. . 3 |- ( ph -> ( x e. A |-> B ) e. (SMblFn ` S ) )
4		eqid 2622	. . 3 |- dom ( x e. A |-> B ) = dom ( x e. A |-> B )
5		smfpimltmpt.r	. . 3 |- ( ph -> R e. RR )
6	1, 2, 3, 4, 5	smfpreimaltf 40945	. 2 |- ( ph -> { x e. dom ( x e. A |-> B ) | ( ( x e. A |-> B ) ` x ) < R } e. ( S ↾t dom ( x e. A |-> B ) ) )
7		smfpimltmpt.x	. . . . . 6 |- F/ x ph
8		eqid 2622	. . . . . 6 |- ( x e. A |-> B ) = ( x e. A |-> B )
9		smfpimltmpt.b	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. V )
10	7, 8, 9	dmmptdf 39417	. . . . 5 |- ( ph -> dom ( x e. A |-> B ) = A )
11	1	nfdm 5367	. . . . . 6 |- F/_ x dom ( x e. A |-> B )
12		nfcv 2764	. . . . . 6 |- F/_ x A
13	11, 12	rabeqf 3190	. . . . 5 |- ( dom ( x e. A |-> B ) = A -> { x e. dom ( x e. A |-> B ) | ( ( x e. A |-> B ) ` x ) < R } = { x e. A | ( ( x e. A |-> B ) ` x ) < R } )
14	10, 13	syl 17	. . . 4 |- ( ph -> { x e. dom ( x e. A |-> B ) | ( ( x e. A |-> B ) ` x ) < R } = { x e. A | ( ( x e. A |-> B ) ` x ) < R } )
15	8	a1i 11	. . . . . . 7 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
16	15, 9	fvmpt2d 6293	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( x e. A |-> B ) ` x ) = B )
17	16	breq1d 4663	. . . . 5 |- ( ( ph /\ x e. A ) -> ( ( ( x e. A |-> B ) ` x ) < R <-> B < R ) )
18	7, 17	rabbida 39274	. . . 4 |- ( ph -> { x e. A | ( ( x e. A |-> B ) ` x ) < R } = { x e. A | B < R } )
19		eqidd 2623	. . . 4 |- ( ph -> { x e. A | B < R } = { x e. A | B < R } )
20	14, 18, 19	3eqtrrd 2661	. . 3 |- ( ph -> { x e. A | B < R } = { x e. dom ( x e. A |-> B ) | ( ( x e. A |-> B ) ` x ) < R } )
21	10	eqcomd 2628	. . . 4 |- ( ph -> A = dom ( x e. A |-> B ) )
22	21	oveq2d 6666	. . 3 |- ( ph -> ( S ↾t A ) = ( S ↾t dom ( x e. A |-> B ) ) )
23	20, 22	eleq12d 2695	. 2 |- ( ph -> ( { x e. A | B < R } e. ( S ↾t A ) <-> { x e. dom ( x e. A |-> B ) | ( ( x e. A |-> B ) ` x ) < R } e. ( S ↾t dom ( x e. A |-> B ) ) ) )
24	6, 23	mpbird 247	1 |- ( ph -> { x e. A | B < R } e. ( S ↾t A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   {crab 2916   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` 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:  smfaddlem2  40972  smfrec  40996