Theorem limsupresxr 39998

Description: The superior limit of a function only depends on the restriction of that function to the preimage of the set of extended reals. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

Hypotheses

Ref	Expression
limsupresxr.1	|- ( ph -> F e. V )
limsupresxr.2	|- ( ph -> Fun F )
limsupresxr.3	|- A = ( `' F " RR* )

Assertion

Ref	Expression
limsupresxr	|- ( ph -> ( limsup ` ( F |` A ) ) = ( limsup ` F ) )

Proof of Theorem limsupresxr

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

Step	Hyp	Ref	Expression
1		resimass 39449	. . . . . . . . 9 |- ( ( F |` A ) " ( k [,) +oo ) ) C_ ( F " ( k [,) +oo ) )
2	1	a1i 11	. . . . . . . 8 |- ( ph -> ( ( F |` A ) " ( k [,) +oo ) ) C_ ( F " ( k [,) +oo ) ) )
3	2	ssrind 39333	. . . . . . 7 |- ( ph -> ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) C_ ( ( F " ( k [,) +oo ) ) i^i RR* ) )
4		limsupresxr.2	. . . . . . . . . . . . 13 |- ( ph -> Fun F )
5	4	funfnd 5919	. . . . . . . . . . . 12 |- ( ph -> F Fn dom F )
6		elinel1 3799	. . . . . . . . . . . 12 |- ( y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) -> y e. ( F " ( k [,) +oo ) ) )
7		fvelima2 39475	. . . . . . . . . . . 12 |- ( ( F Fn dom F /\ y e. ( F " ( k [,) +oo ) ) ) -> E. x e. ( dom F i^i ( k [,) +oo ) ) ( F ` x ) = y )
8	5, 6, 7	syl2an 494	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) -> E. x e. ( dom F i^i ( k [,) +oo ) ) ( F ` x ) = y )
9		elinel1 3799	. . . . . . . . . . . . . . . . . . . . 21 |- ( x e. ( dom F i^i ( k [,) +oo ) ) -> x e. dom F )
10	9	3ad2ant2 1083	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) /\ x e. ( dom F i^i ( k [,) +oo ) ) /\ ( F ` x ) = y ) -> x e. dom F )
11		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) /\ ( F ` x ) = y ) -> ( F ` x ) = y )
12		elinel2 3800	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) -> y e. RR* )
13	12	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) /\ ( F ` x ) = y ) -> y e. RR* )
14	11, 13	eqeltrd 2701	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) /\ ( F ` x ) = y ) -> ( F ` x ) e. RR* )
15	14	3adant2 1080	. . . . . . . . . . . . . . . . . . . 20 |- ( ( y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) /\ x e. ( dom F i^i ( k [,) +oo ) ) /\ ( F ` x ) = y ) -> ( F ` x ) e. RR* )
16	10, 15	jca 554	. . . . . . . . . . . . . . . . . . 19 |- ( ( y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) /\ x e. ( dom F i^i ( k [,) +oo ) ) /\ ( F ` x ) = y ) -> ( x e. dom F /\ ( F ` x ) e. RR* ) )
17	16	3adant1l 1318	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) /\ ( F ` x ) = y ) -> ( x e. dom F /\ ( F ` x ) e. RR* ) )
18		simp1l 1085	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) /\ ( F ` x ) = y ) -> ph )
19		elpreima 6337	. . . . . . . . . . . . . . . . . . . 20 |- ( F Fn dom F -> ( x e. ( `' F " RR* ) <-> ( x e. dom F /\ ( F ` x ) e. RR* ) ) )
20	5, 19	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( x e. ( `' F " RR* ) <-> ( x e. dom F /\ ( F ` x ) e. RR* ) ) )
21	18, 20	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) /\ ( F ` x ) = y ) -> ( x e. ( `' F " RR* ) <-> ( x e. dom F /\ ( F ` x ) e. RR* ) ) )
22	17, 21	mpbird 247	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) /\ ( F ` x ) = y ) -> x e. ( `' F " RR* ) )
23		limsupresxr.3	. . . . . . . . . . . . . . . . 17 |- A = ( `' F " RR* )
24	22, 23	syl6eleqr 2712	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) /\ ( F ` x ) = y ) -> x e. A )
25	24	3expa 1265	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> x e. A )
26	25	fvresd 6208	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> ( ( F |` A ) ` x ) = ( F ` x ) )
27		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> ( F ` x ) = y )
28	26, 27	eqtr2d 2657	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> y = ( ( F |` A ) ` x ) )
29		simplll 798	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> ph )
30	4	funresd 39476	. . . . . . . . . . . . . . . 16 |- ( ph -> Fun ( F |` A ) )
31	29, 30	syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> Fun ( F |` A ) )
32	9	ad2antlr 763	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> x e. dom F )
33	25, 32	elind 3798	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> x e. ( A i^i dom F ) )
34		dmres 5419	. . . . . . . . . . . . . . . 16 |- dom ( F |` A ) = ( A i^i dom F )
35	33, 34	syl6eleqr 2712	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> x e. dom ( F |` A ) )
36	31, 35	jca 554	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> ( Fun ( F |` A ) /\ x e. dom ( F |` A ) ) )
37		elinel2 3800	. . . . . . . . . . . . . . 15 |- ( x e. ( dom F i^i ( k [,) +oo ) ) -> x e. ( k [,) +oo ) )
38	37	ad2antlr 763	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> x e. ( k [,) +oo ) )
39		funfvima 6492	. . . . . . . . . . . . . 14 |- ( ( Fun ( F |` A ) /\ x e. dom ( F |` A ) ) -> ( x e. ( k [,) +oo ) -> ( ( F |` A ) ` x ) e. ( ( F |` A ) " ( k [,) +oo ) ) ) )
40	36, 38, 39	sylc 65	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> ( ( F |` A ) ` x ) e. ( ( F |` A ) " ( k [,) +oo ) ) )
41	28, 40	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) /\ x e. ( dom F i^i ( k [,) +oo ) ) ) /\ ( F ` x ) = y ) -> y e. ( ( F |` A ) " ( k [,) +oo ) ) )
42	41	rexlimdva2 39339	. . . . . . . . . . 11 |- ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) -> ( E. x e. ( dom F i^i ( k [,) +oo ) ) ( F ` x ) = y -> y e. ( ( F |` A ) " ( k [,) +oo ) ) ) )
43	8, 42	mpd 15	. . . . . . . . . 10 |- ( ( ph /\ y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) ) -> y e. ( ( F |` A ) " ( k [,) +oo ) ) )
44	43	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) y e. ( ( F |` A ) " ( k [,) +oo ) ) )
45		dfss3 3592	. . . . . . . . 9 |- ( ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ ( ( F |` A ) " ( k [,) +oo ) ) <-> A. y e. ( ( F " ( k [,) +oo ) ) i^i RR* ) y e. ( ( F |` A ) " ( k [,) +oo ) ) )
46	44, 45	sylibr 224	. . . . . . . 8 |- ( ph -> ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ ( ( F |` A ) " ( k [,) +oo ) ) )
47		inss2 3834	. . . . . . . . 9 |- ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ RR*
48	47	a1i 11	. . . . . . . 8 |- ( ph -> ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ RR* )
49	46, 48	ssind 3837	. . . . . . 7 |- ( ph -> ( ( F " ( k [,) +oo ) ) i^i RR* ) C_ ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) )
50	3, 49	eqssd 3620	. . . . . 6 |- ( ph -> ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) = ( ( F " ( k [,) +oo ) ) i^i RR* ) )
51	50	supeq1d 8352	. . . . 5 |- ( ph -> sup ( ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) = sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
52	51	mpteq2dv 4745	. . . 4 |- ( ph -> ( k e. RR |-> sup ( ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
53	52	rneqd 5353	. . 3 |- ( ph -> ran ( k e. RR |-> sup ( ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) )
54	53	infeq1d 8383	. 2 |- ( ph -> inf ( ran ( k e. RR |-> sup ( ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) = inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
55		limsupresxr.1	. . . 4 |- ( ph -> F e. V )
56	55	resexd 39321	. . 3 |- ( ph -> ( F |` A ) e. _V )
57		eqid 2622	. . . 4 |- ( k e. RR |-> sup ( ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> sup ( ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
58	57	limsupval 14205	. . 3 |- ( ( F |` A ) e. _V -> ( limsup ` ( F |` A ) ) = inf ( ran ( k e. RR |-> sup ( ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
59	56, 58	syl 17	. 2 |- ( ph -> ( limsup ` ( F |` A ) ) = inf ( ran ( k e. RR |-> sup ( ( ( ( F |` A ) " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
60		eqid 2622	. . . 4 |- ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) = ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) )
61	60	limsupval 14205	. . 3 |- ( F e. V -> ( limsup ` F ) = inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
62	55, 61	syl 17	. 2 |- ( ph -> ( limsup ` F ) = inf ( ran ( k e. RR |-> sup ( ( ( F " ( k [,) +oo ) ) i^i RR* ) , RR* , < ) ) , RR* , < ) )
63	54, 59, 62	3eqtr4d 2666	1 |- ( ph -> ( limsup ` ( F |` A ) ) = ( limsup ` F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supcsup 8346  infcinf 8347   RRcr 9935   +oocpnf 10071   RR*cxr 10073   < clt 10074   [,)cico 12177   limsupclsp 14201

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-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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-limsup 14202

This theorem is referenced by: (None)