Theorem smfpimbor1lem2 41006

Description: Given a sigma-measurable function, the preimage of a Borel set belongs to the subspace sigma-algebra induced by the domain of the function. Proposition 121E (f) of [Fremlin1] p. 37 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smfpimbor1lem2.s	|- ( ph -> S e. SAlg )
smfpimbor1lem2.f	|- ( ph -> F e. (SMblFn ` S ) )
smfpimbor1lem2.a	|- D = dom F
smfpimbor1lem2.j	|- J = ( topGen ` ran (,) )
smfpimbor1lem2.b	|- B = (SalGen ` J )
smfpimbor1lem2.e	|- ( ph -> E e. B )
smfpimbor1lem2.p	|- P = ( `' F " E )
smfpimbor1lem2.t	|- T = { e e. ~P RR | ( `' F " e ) e. ( S ↾t D ) }

Assertion

Ref	Expression
smfpimbor1lem2	|- ( ph -> P e. ( S ↾t D ) )

Distinct variable groups:   D, e   e, E   e, F   e, J   S, e   ph, e

Allowed substitution hints:   B( e)   P( e)   T( e)

Proof of Theorem smfpimbor1lem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfpimbor1lem2.p	. 2 |- P = ( `' F " E )
2		smfpimbor1lem2.j	. . . . . . . 8 |- J = ( topGen ` ran (,) )
3		retop 22565	. . . . . . . 8 |- ( topGen ` ran (,) ) e. Top
4	2, 3	eqeltri 2697	. . . . . . 7 |- J e. Top
5	4	a1i 11	. . . . . 6 |- ( ph -> J e. Top )
6		smfpimbor1lem2.b	. . . . . 6 |- B = (SalGen ` J )
7		smfpimbor1lem2.s	. . . . . . 7 |- ( ph -> S e. SAlg )
8		smfpimbor1lem2.f	. . . . . . 7 |- ( ph -> F e. (SMblFn ` S ) )
9		smfpimbor1lem2.a	. . . . . . 7 |- D = dom F
10		smfpimbor1lem2.t	. . . . . . 7 |- T = { e e. ~P RR | ( `' F " e ) e. ( S ↾t D ) }
11	7, 8, 9, 10	smfresal 40995	. . . . . 6 |- ( ph -> T e. SAlg )
12	7	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. J ) -> S e. SAlg )
13	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. J ) -> F e. (SMblFn ` S ) )
14		simpr 477	. . . . . . . 8 |- ( ( ph /\ x e. J ) -> x e. J )
15	12, 13, 9, 2, 14, 10	smfpimbor1lem1 41005	. . . . . . 7 |- ( ( ph /\ x e. J ) -> x e. T )
16	15	ssd 39252	. . . . . 6 |- ( ph -> J C_ T )
17		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ e x
18		nfrab1 3122	. . . . . . . . . . . . . . 15 |- F/_ e { e e. ~P RR | ( `' F " e ) e. ( S ↾t D ) }
19	10, 18	nfcxfr 2762	. . . . . . . . . . . . . 14 |- F/_ e T
20	17, 19	eluni2f 39286	. . . . . . . . . . . . 13 |- ( x e. U. T <-> E. e e. T x e. e )
21	20	biimpi 206	. . . . . . . . . . . 12 |- ( x e. U. T -> E. e e. T x e. e )
22	19	nfuni 4442	. . . . . . . . . . . . . 14 |- F/_ e U. T
23	17, 22	nfel 2777	. . . . . . . . . . . . 13 |- F/ e x e. U. T
24		nfv 1843	. . . . . . . . . . . . 13 |- F/ e x e. RR
25	10	eleq2i 2693	. . . . . . . . . . . . . . . . . . . 20 |- ( e e. T <-> e e. { e e. ~P RR | ( `' F " e ) e. ( S ↾t D ) } )
26	25	biimpi 206	. . . . . . . . . . . . . . . . . . 19 |- ( e e. T -> e e. { e e. ~P RR | ( `' F " e ) e. ( S ↾t D ) } )
27		rabidim1 3117	. . . . . . . . . . . . . . . . . . 19 |- ( e e. { e e. ~P RR | ( `' F " e ) e. ( S ↾t D ) } -> e e. ~P RR )
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( e e. T -> e e. ~P RR )
29		elpwi 4168	. . . . . . . . . . . . . . . . . 18 |- ( e e. ~P RR -> e C_ RR )
30	28, 29	syl 17	. . . . . . . . . . . . . . . . 17 |- ( e e. T -> e C_ RR )
31	30	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( e e. T /\ x e. e ) -> e C_ RR )
32		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( e e. T /\ x e. e ) -> x e. e )
33	31, 32	sseldd 3604	. . . . . . . . . . . . . . 15 |- ( ( e e. T /\ x e. e ) -> x e. RR )
34	33	ex 450	. . . . . . . . . . . . . 14 |- ( e e. T -> ( x e. e -> x e. RR ) )
35	34	a1i 11	. . . . . . . . . . . . 13 |- ( x e. U. T -> ( e e. T -> ( x e. e -> x e. RR ) ) )
36	23, 24, 35	rexlimd 3026	. . . . . . . . . . . 12 |- ( x e. U. T -> ( E. e e. T x e. e -> x e. RR ) )
37	21, 36	mpd 15	. . . . . . . . . . 11 |- ( x e. U. T -> x e. RR )
38	37	rgen 2922	. . . . . . . . . 10 |- A. x e. U. T x e. RR
39		dfss3 3592	. . . . . . . . . 10 |- ( U. T C_ RR <-> A. x e. U. T x e. RR )
40	38, 39	mpbir 221	. . . . . . . . 9 |- U. T C_ RR
41	40	a1i 11	. . . . . . . 8 |- ( ph -> U. T C_ RR )
42		uniretop 22566	. . . . . . . . . . . 12 |- RR = U. ( topGen ` ran (,) )
43	2	eqcomi 2631	. . . . . . . . . . . . 13 |- ( topGen ` ran (,) ) = J
44	43	unieqi 4445	. . . . . . . . . . . 12 |- U. ( topGen ` ran (,) ) = U. J
45	42, 44	eqtr2i 2645	. . . . . . . . . . 11 |- U. J = RR
46	45	a1i 11	. . . . . . . . . 10 |- ( ph -> U. J = RR )
47	46	eqcomd 2628	. . . . . . . . 9 |- ( ph -> RR = U. J )
48	16	unissd 4462	. . . . . . . . 9 |- ( ph -> U. J C_ U. T )
49	47, 48	eqsstrd 3639	. . . . . . . 8 |- ( ph -> RR C_ U. T )
50	41, 49	eqssd 3620	. . . . . . 7 |- ( ph -> U. T = RR )
51	50, 46	eqtr4d 2659	. . . . . 6 |- ( ph -> U. T = U. J )
52	5, 6, 11, 16, 51	salgenss 40554	. . . . 5 |- ( ph -> B C_ T )
53		smfpimbor1lem2.e	. . . . 5 |- ( ph -> E e. B )
54	52, 53	sseldd 3604	. . . 4 |- ( ph -> E e. T )
55		imaeq2 5462	. . . . . 6 |- ( e = E -> ( `' F " e ) = ( `' F " E ) )
56	55	eleq1d 2686	. . . . 5 |- ( e = E -> ( ( `' F " e ) e. ( S ↾t D ) <-> ( `' F " E ) e. ( S ↾t D ) ) )
57	56, 10	elrab2 3366	. . . 4 |- ( E e. T <-> ( E e. ~P RR /\ ( `' F " E ) e. ( S ↾t D ) ) )
58	54, 57	sylib 208	. . 3 |- ( ph -> ( E e. ~P RR /\ ( `' F " E ) e. ( S ↾t D ) ) )
59	58	simprd 479	. 2 |- ( ph -> ( `' F " E ) e. ( S ↾t D ) )
60	1, 59	syl5eqel 2705	1 |- ( ph -> P e. ( S ↾t D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   ` cfv 5888  (class class class)co 6650   RRcr 9935   (,)cioo 12175   ↾_t crest 16081   topGenctg 16098   Topctop 20698  SAlgcsalg 40528  SalGencsalgen 40532  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-omul 7565  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-oi 8415  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-topgen 16104  df-top 20699  df-bases 20750  df-salg 40529  df-salgen 40533  df-smblfn 40910

This theorem is referenced by:  smfpimbor1  41007