Theorem smfpimcc 41014

Description: Given a countable set of sigma-measurable functions, and a Borel set A there exists a choice function h that, for each measurable function, chooses a measurable set that, when intersected with the function's domain, gives the preimage of A. This is a generalization of the observation at the beginning of the proof of Proposition 121F of [Fremlin1] p. 39 . The statement would also be provable for uncountable sets, but in most cases it will suffice to consider the countable case, and only the axiom of countable choice will be needed. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smfpimcc.1	|- F/_ n F
smfpimcc.z	|- Z = ( ZZ>= ` M )
smfpimcc.s	|- ( ph -> S e. SAlg )
smfpimcc.f	|- ( ph -> F : Z --> (SMblFn ` S ) )
smfpimcc.j	|- J = ( topGen ` ran (,) )
smfpimcc.b	|- B = (SalGen ` J )
smfpimcc.a	|- ( ph -> A e. B )

Assertion

Ref	Expression
smfpimcc	|- ( ph -> E. h ( h : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) )

Distinct variable groups:   A, h, n   h, F   S, h   h, Z, n

Allowed substitution hints:   ph( h, n)   B( h, n)   S( n)   F( n)   J( h, n)   M( h, n)

Proof of Theorem smfpimcc

Dummy variables f m s w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfpimcc.z	. . . . . . 7 |- Z = ( ZZ>= ` M )
2	1	uzct 39232	. . . . . 6 |- Z ~<_ om
3	2	a1i 11	. . . . 5 |- ( ph -> Z ~<_ om )
4		mptct 9360	. . . . 5 |- ( Z ~<_ om -> ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ~<_ om )
5		rnct 9347	. . . . 5 |- ( ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ~<_ om -> ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ~<_ om )
6	3, 4, 5	3syl 18	. . . 4 |- ( ph -> ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ~<_ om )
7		vex 3203	. . . . . . . 8 |- y e. _V
8		eqid 2622	. . . . . . . . 9 |- ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) = ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } )
9	8	elrnmpt 5372	. . . . . . . 8 |- ( y e. _V -> ( y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) <-> E. m e. Z y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) )
10	7, 9	ax-mp 5	. . . . . . 7 |- ( y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) <-> E. m e. Z y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } )
11	10	biimpi 206	. . . . . 6 |- ( y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) -> E. m e. Z y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } )
12	11	adantl 482	. . . . 5 |- ( ( ph /\ y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ) -> E. m e. Z y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } )
13		simp3 1063	. . . . . . . . 9 |- ( ( ph /\ m e. Z /\ y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) -> y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } )
14		smfpimcc.s	. . . . . . . . . . . . . 14 |- ( ph -> S e. SAlg )
15	14	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. Z ) -> S e. SAlg )
16		smfpimcc.f	. . . . . . . . . . . . . 14 |- ( ph -> F : Z --> (SMblFn ` S ) )
17	16	ffvelrnda 6359	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. Z ) -> ( F ` m ) e. (SMblFn ` S ) )
18		eqid 2622	. . . . . . . . . . . . 13 |- dom ( F ` m ) = dom ( F ` m )
19		smfpimcc.j	. . . . . . . . . . . . 13 |- J = ( topGen ` ran (,) )
20		smfpimcc.b	. . . . . . . . . . . . 13 |- B = (SalGen ` J )
21		smfpimcc.a	. . . . . . . . . . . . . 14 |- ( ph -> A e. B )
22	21	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ m e. Z ) -> A e. B )
23		eqid 2622	. . . . . . . . . . . . 13 |- ( `' ( F ` m ) " A ) = ( `' ( F ` m ) " A )
24	15, 17, 18, 19, 20, 22, 23	smfpimbor1 41007	. . . . . . . . . . . 12 |- ( ( ph /\ m e. Z ) -> ( `' ( F ` m ) " A ) e. ( S ↾t dom ( F ` m ) ) )
25		fvex 6201	. . . . . . . . . . . . . . . 16 |- ( F ` m ) e. _V
26	25	dmex 7099	. . . . . . . . . . . . . . 15 |- dom ( F ` m ) e. _V
27	26	a1i 11	. . . . . . . . . . . . . 14 |- ( ph -> dom ( F ` m ) e. _V )
28		elrest 16088	. . . . . . . . . . . . . 14 |- ( ( S e. SAlg /\ dom ( F ` m ) e. _V ) -> ( ( `' ( F ` m ) " A ) e. ( S ↾t dom ( F ` m ) ) <-> E. s e. S ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) ) )
29	14, 27, 28	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( ( `' ( F ` m ) " A ) e. ( S ↾t dom ( F ` m ) ) <-> E. s e. S ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) ) )
30	29	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ m e. Z ) -> ( ( `' ( F ` m ) " A ) e. ( S ↾t dom ( F ` m ) ) <-> E. s e. S ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) ) )
31	24, 30	mpbid 222	. . . . . . . . . . 11 |- ( ( ph /\ m e. Z ) -> E. s e. S ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) )
32		rabn0 3958	. . . . . . . . . . 11 |- ( { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } =/= (/) <-> E. s e. S ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) )
33	31, 32	sylibr 224	. . . . . . . . . 10 |- ( ( ph /\ m e. Z ) -> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } =/= (/) )
34	33	3adant3 1081	. . . . . . . . 9 |- ( ( ph /\ m e. Z /\ y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) -> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } =/= (/) )
35	13, 34	eqnetrd 2861	. . . . . . . 8 |- ( ( ph /\ m e. Z /\ y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) -> y =/= (/) )
36	35	3exp 1264	. . . . . . 7 |- ( ph -> ( m e. Z -> ( y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } -> y =/= (/) ) ) )
37	36	rexlimdv 3030	. . . . . 6 |- ( ph -> ( E. m e. Z y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } -> y =/= (/) ) )
38	37	adantr 481	. . . . 5 |- ( ( ph /\ y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ) -> ( E. m e. Z y = { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } -> y =/= (/) ) )
39	12, 38	mpd 15	. . . 4 |- ( ( ph /\ y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ) -> y =/= (/) )
40	6, 39	axccd2 39430	. . 3 |- ( ph -> E. f A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y )
41		nfv 1843	. . . . . . 7 |- F/ m ph
42		nfmpt1 4747	. . . . . . . . 9 |- F/_ m ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } )
43	42	nfrn 5368	. . . . . . . 8 |- F/_ m ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } )
44		nfv 1843	. . . . . . . 8 |- F/ m ( f ` y ) e. y
45	43, 44	nfral 2945	. . . . . . 7 |- F/ m A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y
46	41, 45	nfan 1828	. . . . . 6 |- F/ m ( ph /\ A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y )
47		fvex 6201	. . . . . . 7 |- ( ZZ>= ` M ) e. _V
48	1, 47	eqeltri 2697	. . . . . 6 |- Z e. _V
49	14	adantr 481	. . . . . 6 |- ( ( ph /\ A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y ) -> S e. SAlg )
50		fveq2 6191	. . . . . . . . 9 |- ( y = w -> ( f ` y ) = ( f ` w ) )
51		id 22	. . . . . . . . 9 |- ( y = w -> y = w )
52	50, 51	eleq12d 2695	. . . . . . . 8 |- ( y = w -> ( ( f ` y ) e. y <-> ( f ` w ) e. w ) )
53	52	rspccva 3308	. . . . . . 7 |- ( ( A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y /\ w e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ) -> ( f ` w ) e. w )
54	53	adantll 750	. . . . . 6 |- ( ( ( ph /\ A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y ) /\ w e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ) -> ( f ` w ) e. w )
55		eqid 2622	. . . . . 6 |- ( m e. Z |-> ( f ` { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ) = ( m e. Z |-> ( f ` { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) )
56	46, 48, 49, 54, 55	smfpimcclem 41013	. . . . 5 |- ( ( ph /\ A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y ) -> E. h ( h : Z --> S /\ A. m e. Z ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) ) ) )
57	56	ex 450	. . . 4 |- ( ph -> ( A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y -> E. h ( h : Z --> S /\ A. m e. Z ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) ) ) ) )
58	57	exlimdv 1861	. . 3 |- ( ph -> ( E. f A. y e. ran ( m e. Z |-> { s e. S | ( `' ( F ` m ) " A ) = ( s i^i dom ( F ` m ) ) } ) ( f ` y ) e. y -> E. h ( h : Z --> S /\ A. m e. Z ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) ) ) ) )
59	40, 58	mpd 15	. 2 |- ( ph -> E. h ( h : Z --> S /\ A. m e. Z ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) ) ) )
60		smfpimcc.1	. . . . . . . . 9 |- F/_ n F
61		nfcv 2764	. . . . . . . . 9 |- F/_ n m
62	60, 61	nffv 6198	. . . . . . . 8 |- F/_ n ( F ` m )
63	62	nfcnv 5301	. . . . . . 7 |- F/_ n `' ( F ` m )
64		nfcv 2764	. . . . . . 7 |- F/_ n A
65	63, 64	nfima 5474	. . . . . 6 |- F/_ n ( `' ( F ` m ) " A )
66		nfcv 2764	. . . . . . 7 |- F/_ n ( h ` m )
67	62	nfdm 5367	. . . . . . 7 |- F/_ n dom ( F ` m )
68	66, 67	nfin 3820	. . . . . 6 |- F/_ n ( ( h ` m ) i^i dom ( F ` m ) )
69	65, 68	nfeq 2776	. . . . 5 |- F/ n ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) )
70		nfv 1843	. . . . 5 |- F/ m ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) )
71		fveq2 6191	. . . . . . . 8 |- ( m = n -> ( F ` m ) = ( F ` n ) )
72	71	cnveqd 5298	. . . . . . 7 |- ( m = n -> `' ( F ` m ) = `' ( F ` n ) )
73	72	imaeq1d 5465	. . . . . 6 |- ( m = n -> ( `' ( F ` m ) " A ) = ( `' ( F ` n ) " A ) )
74		fveq2 6191	. . . . . . 7 |- ( m = n -> ( h ` m ) = ( h ` n ) )
75	71	dmeqd 5326	. . . . . . 7 |- ( m = n -> dom ( F ` m ) = dom ( F ` n ) )
76	74, 75	ineq12d 3815	. . . . . 6 |- ( m = n -> ( ( h ` m ) i^i dom ( F ` m ) ) = ( ( h ` n ) i^i dom ( F ` n ) ) )
77	73, 76	eqeq12d 2637	. . . . 5 |- ( m = n -> ( ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) ) <-> ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) )
78	69, 70, 77	cbvral 3167	. . . 4 |- ( A. m e. Z ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) ) <-> A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) )
79	78	anbi2i 730	. . 3 |- ( ( h : Z --> S /\ A. m e. Z ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) ) ) <-> ( h : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) )
80	79	exbii 1774	. 2 |- ( E. h ( h : Z --> S /\ A. m e. Z ( `' ( F ` m ) " A ) = ( ( h ` m ) i^i dom ( F ` m ) ) ) <-> E. h ( h : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) )
81	59, 80	sylib 208	1 |- ( ph -> E. h ( h : Z --> S /\ A. n e. Z ( `' ( F ` n ) " A ) = ( ( h ` n ) i^i dom ( F ` n ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   (/)c0 3915   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~<_ cdom 7953   ZZ>=cuz 11687   (,)cioo 12175   ↾_t crest 16081   topGenctg 16098  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:  smfsuplem2  41018