Theorem smflim2 41012

Description: The limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). TODO this has less distinct variable restrictions than smflim and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smflim2.n	|- F/_ m F
smflim2.x	|- F/_ x F
smflim2.m	|- ( ph -> M e. ZZ )
smflim2.z	|- Z = ( ZZ>= ` M )
smflim2.s	|- ( ph -> S e. SAlg )
smflim2.f	|- ( ph -> F : Z --> (SMblFn ` S ) )
smflim2.d	|- D = { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( m e. Z |-> ( ( F ` m ) ` x ) ) e. dom ~~> }
smflim2.g	|- G = ( x e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) )

Assertion

Ref	Expression
smflim2	|- ( ph -> G e. (SMblFn ` S ) )

Distinct variable groups:   n, F   m, Z, n, x

Allowed substitution hints:   ph( x, m, n)   D( x, m, n)   S( x, m, n)   F( x, m)   G( x, m, n)   M( x, m, n)

Proof of Theorem smflim2

Dummy variables k y j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ j F
2		nfcv 2764	. 2 |- F/_ y F
3		smflim2.m	. 2 |- ( ph -> M e. ZZ )
4		smflim2.z	. 2 |- Z = ( ZZ>= ` M )
5		smflim2.s	. 2 |- ( ph -> S e. SAlg )
6		smflim2.f	. 2 |- ( ph -> F : Z --> (SMblFn ` S ) )
7		smflim2.d	. . 3 |- D = { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( m e. Z |-> ( ( F ` m ) ` x ) ) e. dom ~~> }
8		nfcv 2764	. . . . 5 |- F/_ x Z
9		nfcv 2764	. . . . . 6 |- F/_ x ( ZZ>= ` n )
10		smflim2.x	. . . . . . . 8 |- F/_ x F
11		nfcv 2764	. . . . . . . 8 |- F/_ x m
12	10, 11	nffv 6198	. . . . . . 7 |- F/_ x ( F ` m )
13	12	nfdm 5367	. . . . . 6 |- F/_ x dom ( F ` m )
14	9, 13	nfiin 4549	. . . . 5 |- F/_ x |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
15	8, 14	nfiun 4548	. . . 4 |- F/_ x U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
16		nfcv 2764	. . . 4 |- F/_ y U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m )
17		nfv 1843	. . . 4 |- F/ y ( m e. Z |-> ( ( F ` m ) ` x ) ) e. dom ~~>
18		nfcv 2764	. . . . . . 7 |- F/_ j ( ( F ` m ) ` y )
19		smflim2.n	. . . . . . . . 9 |- F/_ m F
20		nfcv 2764	. . . . . . . . 9 |- F/_ m j
21	19, 20	nffv 6198	. . . . . . . 8 |- F/_ m ( F ` j )
22		nfcv 2764	. . . . . . . 8 |- F/_ m y
23	21, 22	nffv 6198	. . . . . . 7 |- F/_ m ( ( F ` j ) ` y )
24		fveq2 6191	. . . . . . . 8 |- ( m = j -> ( F ` m ) = ( F ` j ) )
25	24	fveq1d 6193	. . . . . . 7 |- ( m = j -> ( ( F ` m ) ` y ) = ( ( F ` j ) ` y ) )
26	18, 23, 25	cbvmpt 4749	. . . . . 6 |- ( m e. Z |-> ( ( F ` m ) ` y ) ) = ( j e. Z |-> ( ( F ` j ) ` y ) )
27		nfcv 2764	. . . . . . . . 9 |- F/_ x j
28	10, 27	nffv 6198	. . . . . . . 8 |- F/_ x ( F ` j )
29		nfcv 2764	. . . . . . . 8 |- F/_ x y
30	28, 29	nffv 6198	. . . . . . 7 |- F/_ x ( ( F ` j ) ` y )
31	8, 30	nfmpt 4746	. . . . . 6 |- F/_ x ( j e. Z |-> ( ( F ` j ) ` y ) )
32	26, 31	nfcxfr 2762	. . . . 5 |- F/_ x ( m e. Z |-> ( ( F ` m ) ` y ) )
33		nfcv 2764	. . . . 5 |- F/_ x dom ~~>
34	32, 33	nfel 2777	. . . 4 |- F/ x ( m e. Z |-> ( ( F ` m ) ` y ) ) e. dom ~~>
35		fveq2 6191	. . . . . 6 |- ( x = y -> ( ( F ` m ) ` x ) = ( ( F ` m ) ` y ) )
36	35	mpteq2dv 4745	. . . . 5 |- ( x = y -> ( m e. Z |-> ( ( F ` m ) ` x ) ) = ( m e. Z |-> ( ( F ` m ) ` y ) ) )
37	36	eleq1d 2686	. . . 4 |- ( x = y -> ( ( m e. Z |-> ( ( F ` m ) ` x ) ) e. dom ~~> <-> ( m e. Z |-> ( ( F ` m ) ` y ) ) e. dom ~~> ) )
38	15, 16, 17, 34, 37	cbvrab 3198	. . 3 |- { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( m e. Z |-> ( ( F ` m ) ` x ) ) e. dom ~~> } = { y e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( m e. Z |-> ( ( F ` m ) ` y ) ) e. dom ~~> }
39		fveq2 6191	. . . . . . . . 9 |- ( n = k -> ( ZZ>= ` n ) = ( ZZ>= ` k ) )
40	39	iineq1d 39267	. . . . . . . 8 |- ( n = k -> |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = |^|_ m e. ( ZZ>= ` k ) dom ( F ` m ) )
41		nfcv 2764	. . . . . . . . . 10 |- F/_ j dom ( F ` m )
42	21	nfdm 5367	. . . . . . . . . 10 |- F/_ m dom ( F ` j )
43	24	dmeqd 5326	. . . . . . . . . 10 |- ( m = j -> dom ( F ` m ) = dom ( F ` j ) )
44	41, 42, 43	cbviin 4558	. . . . . . . . 9 |- |^|_ m e. ( ZZ>= ` k ) dom ( F ` m ) = |^|_ j e. ( ZZ>= ` k ) dom ( F ` j )
45	44	a1i 11	. . . . . . . 8 |- ( n = k -> |^|_ m e. ( ZZ>= ` k ) dom ( F ` m ) = |^|_ j e. ( ZZ>= ` k ) dom ( F ` j ) )
46	40, 45	eqtrd 2656	. . . . . . 7 |- ( n = k -> |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = |^|_ j e. ( ZZ>= ` k ) dom ( F ` j ) )
47	46	cbviunv 4559	. . . . . 6 |- U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) = U_ k e. Z |^|_ j e. ( ZZ>= ` k ) dom ( F ` j )
48	47	eleq2i 2693	. . . . 5 |- ( y e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) <-> y e. U_ k e. Z |^|_ j e. ( ZZ>= ` k ) dom ( F ` j ) )
49	26	eleq1i 2692	. . . . 5 |- ( ( m e. Z |-> ( ( F ` m ) ` y ) ) e. dom ~~> <-> ( j e. Z |-> ( ( F ` j ) ` y ) ) e. dom ~~> )
50	48, 49	anbi12i 733	. . . 4 |- ( ( y e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) /\ ( m e. Z |-> ( ( F ` m ) ` y ) ) e. dom ~~> ) <-> ( y e. U_ k e. Z |^|_ j e. ( ZZ>= ` k ) dom ( F ` j ) /\ ( j e. Z |-> ( ( F ` j ) ` y ) ) e. dom ~~> ) )
51	50	rabbia2 3187	. . 3 |- { y e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( m e. Z |-> ( ( F ` m ) ` y ) ) e. dom ~~> } = { y e. U_ k e. Z |^|_ j e. ( ZZ>= ` k ) dom ( F ` j ) | ( j e. Z |-> ( ( F ` j ) ` y ) ) e. dom ~~> }
52	7, 38, 51	3eqtri 2648	. 2 |- D = { y e. U_ k e. Z |^|_ j e. ( ZZ>= ` k ) dom ( F ` j ) | ( j e. Z |-> ( ( F ` j ) ` y ) ) e. dom ~~> }
53		smflim2.g	. . 3 |- G = ( x e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) )
54		nfrab1 3122	. . . . 5 |- F/_ x { x e. U_ n e. Z |^|_ m e. ( ZZ>= ` n ) dom ( F ` m ) | ( m e. Z |-> ( ( F ` m ) ` x ) ) e. dom ~~> }
55	7, 54	nfcxfr 2762	. . . 4 |- F/_ x D
56		nfcv 2764	. . . 4 |- F/_ y D
57		nfcv 2764	. . . 4 |- F/_ y ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) )
58		nfcv 2764	. . . . 5 |- F/_ x ~~>
59	58, 31	nffv 6198	. . . 4 |- F/_ x ( ~~> ` ( j e. Z |-> ( ( F ` j ) ` y ) ) )
60	26	a1i 11	. . . . . 6 |- ( x = y -> ( m e. Z |-> ( ( F ` m ) ` y ) ) = ( j e. Z |-> ( ( F ` j ) ` y ) ) )
61	36, 60	eqtrd 2656	. . . . 5 |- ( x = y -> ( m e. Z |-> ( ( F ` m ) ` x ) ) = ( j e. Z |-> ( ( F ` j ) ` y ) ) )
62	61	fveq2d 6195	. . . 4 |- ( x = y -> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) = ( ~~> ` ( j e. Z |-> ( ( F ` j ) ` y ) ) ) )
63	55, 56, 57, 59, 62	cbvmptf 4748	. . 3 |- ( x e. D |-> ( ~~> ` ( m e. Z |-> ( ( F ` m ) ` x ) ) ) ) = ( y e. D |-> ( ~~> ` ( j e. Z |-> ( ( F ` j ) ` y ) ) ) )
64	53, 63	eqtri 2644	. 2 |- G = ( y e. D |-> ( ~~> ` ( j e. Z |-> ( ( F ` j ) ` y ) ) ) )
65	1, 2, 3, 4, 5, 6, 52, 64	smflim 40985	1 |- ( ph -> G e. (SMblFn ` S ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   F/_wnfc 2751   {crab 2916   U_ciun 4520   |^|_ciin 4521   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888   ZZcz 11377   ZZ>=cuz 11687   ~~> cli 14215  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-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-2 11079  df-3 11080  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-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-rest 16083  df-salg 40529  df-smblfn 40910

This theorem is referenced by:  smflimmpt  41016