Theorem lmff 21105

Description: If F converges, there is some upper integer set on which F is a total function. (Contributed by Mario Carneiro, 31-Dec-2013.)

Hypotheses

Ref	Expression
lmff.1	|- Z = ( ZZ>= ` M )
lmff.3	|- ( ph -> J e. (TopOn ` X ) )
lmff.4	|- ( ph -> M e. ZZ )
lmff.5	|- ( ph -> F e. dom ( ~~> t ` J ) )

Assertion

Ref	Expression
lmff	|- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )

Distinct variable groups:   j, F   j, J   j, M   ph, j   j, X   j, Z

Proof of Theorem lmff

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

Step	Hyp	Ref	Expression
1		lmff.5	. . . . . 6 |- ( ph -> F e. dom ( ~~> t ` J ) )
2		eldm2g 5320	. . . . . . 7 |- ( F e. dom ( ~~> t ` J ) -> ( F e. dom ( ~~> t ` J ) <-> E. y <. F , y >. e. ( ~~> t ` J ) ) )
3	2	ibi 256	. . . . . 6 |- ( F e. dom ( ~~> t ` J ) -> E. y <. F , y >. e. ( ~~> t ` J ) )
4	1, 3	syl 17	. . . . 5 |- ( ph -> E. y <. F , y >. e. ( ~~> t ` J ) )
5		df-br 4654	. . . . . 6 |- ( F ( ~~> t ` J ) y <-> <. F , y >. e. ( ~~> t ` J ) )
6	5	exbii 1774	. . . . 5 |- ( E. y F ( ~~> t ` J ) y <-> E. y <. F , y >. e. ( ~~> t ` J ) )
7	4, 6	sylibr 224	. . . 4 |- ( ph -> E. y F ( ~~> t ` J ) y )
8		lmff.3	. . . . . . 7 |- ( ph -> J e. (TopOn ` X ) )
9		toponmax 20730	. . . . . . 7 |- ( J e. (TopOn ` X ) -> X e. J )
10	8, 9	syl 17	. . . . . 6 |- ( ph -> X e. J )
11	10	adantr 481	. . . . 5 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> X e. J )
12	8	lmbr 21062	. . . . . . 7 |- ( ph -> ( F ( ~~> t ` J ) y <-> ( F e. ( X ^pm CC ) /\ y e. X /\ A. j e. J ( y e. j -> E. x e. ran ZZ>= ( F |` x ) : x --> j ) ) ) )
13	12	biimpa 501	. . . . . 6 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> ( F e. ( X ^pm CC ) /\ y e. X /\ A. j e. J ( y e. j -> E. x e. ran ZZ>= ( F |` x ) : x --> j ) ) )
14	13	simp3d 1075	. . . . 5 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> A. j e. J ( y e. j -> E. x e. ran ZZ>= ( F |` x ) : x --> j ) )
15		lmcl 21101	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) y ) -> y e. X )
16	8, 15	sylan 488	. . . . 5 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> y e. X )
17		eleq2 2690	. . . . . . 7 |- ( j = X -> ( y e. j <-> y e. X ) )
18		feq3 6028	. . . . . . . 8 |- ( j = X -> ( ( F |` x ) : x --> j <-> ( F |` x ) : x --> X ) )
19	18	rexbidv 3052	. . . . . . 7 |- ( j = X -> ( E. x e. ran ZZ>= ( F |` x ) : x --> j <-> E. x e. ran ZZ>= ( F |` x ) : x --> X ) )
20	17, 19	imbi12d 334	. . . . . 6 |- ( j = X -> ( ( y e. j -> E. x e. ran ZZ>= ( F |` x ) : x --> j ) <-> ( y e. X -> E. x e. ran ZZ>= ( F |` x ) : x --> X ) ) )
21	20	rspcv 3305	. . . . 5 |- ( X e. J -> ( A. j e. J ( y e. j -> E. x e. ran ZZ>= ( F |` x ) : x --> j ) -> ( y e. X -> E. x e. ran ZZ>= ( F |` x ) : x --> X ) ) )
22	11, 14, 16, 21	syl3c 66	. . . 4 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> E. x e. ran ZZ>= ( F |` x ) : x --> X )
23	7, 22	exlimddv 1863	. . 3 |- ( ph -> E. x e. ran ZZ>= ( F |` x ) : x --> X )
24		uzf 11690	. . . 4 |- ZZ>= : ZZ --> ~P ZZ
25		ffn 6045	. . . 4 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
26		reseq2 5391	. . . . . 6 |- ( x = ( ZZ>= ` j ) -> ( F |` x ) = ( F |` ( ZZ>= ` j ) ) )
27		id 22	. . . . . 6 |- ( x = ( ZZ>= ` j ) -> x = ( ZZ>= ` j ) )
28	26, 27	feq12d 6033	. . . . 5 |- ( x = ( ZZ>= ` j ) -> ( ( F |` x ) : x --> X <-> ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) )
29	28	rexrn 6361	. . . 4 |- ( ZZ>= Fn ZZ -> ( E. x e. ran ZZ>= ( F |` x ) : x --> X <-> E. j e. ZZ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) )
30	24, 25, 29	mp2b 10	. . 3 |- ( E. x e. ran ZZ>= ( F |` x ) : x --> X <-> E. j e. ZZ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )
31	23, 30	sylib 208	. 2 |- ( ph -> E. j e. ZZ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )
32		lmff.4	. . . 4 |- ( ph -> M e. ZZ )
33		lmff.1	. . . . 5 |- Z = ( ZZ>= ` M )
34	33	rexuz3 14088	. . . 4 |- ( M e. ZZ -> ( E. j e. Z A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) <-> E. j e. ZZ A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) ) )
35	32, 34	syl 17	. . 3 |- ( ph -> ( E. j e. Z A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) <-> E. j e. ZZ A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) ) )
36	13	simp1d 1073	. . . . . . 7 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> F e. ( X ^pm CC ) )
37	7, 36	exlimddv 1863	. . . . . 6 |- ( ph -> F e. ( X ^pm CC ) )
38		pmfun 7877	. . . . . 6 |- ( F e. ( X ^pm CC ) -> Fun F )
39	37, 38	syl 17	. . . . 5 |- ( ph -> Fun F )
40		ffvresb 6394	. . . . 5 |- ( Fun F -> ( ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X <-> A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) ) )
41	39, 40	syl 17	. . . 4 |- ( ph -> ( ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X <-> A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) ) )
42	41	rexbidv 3052	. . 3 |- ( ph -> ( E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X <-> E. j e. Z A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) ) )
43	41	rexbidv 3052	. . 3 |- ( ph -> ( E. j e. ZZ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X <-> E. j e. ZZ A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) ) )
44	35, 42, 43	3bitr4d 300	. 2 |- ( ph -> ( E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X <-> E. j e. ZZ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) )
45	31, 44	mpbird 247	1 |- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   ~Pcpw 4158   <.cop 4183   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   ZZcz 11377   ZZ>=cuz 11687  TopOnctopon 20715   ~~> tclm 21030

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-neg 10269  df-z 11378  df-uz 11688  df-top 20699  df-topon 20716  df-lm 21033

This theorem is referenced by:  lmle  23099