Theorem lmff 14663

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 4873	. . . . . . 7 |- ( F e. dom ( ~~> t ` J ) -> ( F e. dom ( ~~> t ` J ) <-> E. y <. F , y >. e. ( ~~> t ` J ) ) )
3	2	ibi 176	. . . . . 6 |- ( F e. dom ( ~~> t ` J ) -> E. y <. F , y >. e. ( ~~> t ` J ) )
4	1, 3	syl 14	. . . . 5 |- ( ph -> E. y <. F , y >. e. ( ~~> t ` J ) )
5		df-br 4044	. . . . . 6 |- ( F ( ~~> t ` J ) y <-> <. F , y >. e. ( ~~> t ` J ) )
6	5	exbii 1627	. . . . 5 |- ( E. y F ( ~~> t ` J ) y <-> E. y <. F , y >. e. ( ~~> t ` J ) )
7	4, 6	sylibr 134	. . . 4 |- ( ph -> E. y F ( ~~> t ` J ) y )
8		lmff.3	. . . . . 6 |- ( ph -> J e. (TopOn ` X ) )
9		lmcl 14659	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) y ) -> y e. X )
10	8, 9	sylan 283	. . . . 5 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> y e. X )
11		eleq2 2268	. . . . . . 7 |- ( j = X -> ( y e. j <-> y e. X ) )
12		feq3 5409	. . . . . . . 8 |- ( j = X -> ( ( F |` x ) : x --> j <-> ( F |` x ) : x --> X ) )
13	12	rexbidv 2506	. . . . . . 7 |- ( j = X -> ( E. x e. ran ZZ>= ( F |` x ) : x --> j <-> E. x e. ran ZZ>= ( F |` x ) : x --> X ) )
14	11, 13	imbi12d 234	. . . . . 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 ) ) )
15	8	lmbr 14627	. . . . . . . 8 |- ( 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 ) ) ) )
16	15	biimpa 296	. . . . . . 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 ) ) )
17	16	simp3d 1013	. . . . . 6 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> A. j e. J ( y e. j -> E. x e. ran ZZ>= ( F |` x ) : x --> j ) )
18		toponmax 14439	. . . . . . . 8 |- ( J e. (TopOn ` X ) -> X e. J )
19	8, 18	syl 14	. . . . . . 7 |- ( ph -> X e. J )
20	19	adantr 276	. . . . . 6 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> X e. J )
21	14, 17, 20	rspcdva 2881	. . . . 5 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> ( y e. X -> E. x e. ran ZZ>= ( F |` x ) : x --> X ) )
22	10, 21	mpd 13	. . . 4 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> E. x e. ran ZZ>= ( F |` x ) : x --> X )
23	7, 22	exlimddv 1921	. . 3 |- ( ph -> E. x e. ran ZZ>= ( F |` x ) : x --> X )
24		uzf 9650	. . . 4 |- ZZ>= : ZZ --> ~P ZZ
25		ffn 5424	. . . 4 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
26		reseq2 4953	. . . . . 6 |- ( x = ( ZZ>= ` j ) -> ( F |` x ) = ( F |` ( ZZ>= ` j ) ) )
27		id 19	. . . . . 6 |- ( x = ( ZZ>= ` j ) -> x = ( ZZ>= ` j ) )
28	26, 27	feq12d 5414	. . . . 5 |- ( x = ( ZZ>= ` j ) -> ( ( F |` x ) : x --> X <-> ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) )
29	28	rexrn 5716	. . . 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 8	. . 3 |- ( E. x e. ran ZZ>= ( F |` x ) : x --> X <-> E. j e. ZZ ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )
31	23, 30	sylib 122	. 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 11243	. . . 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 14	. . 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	16	simp1d 1011	. . . . . . 7 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> F e. ( X ^pm CC ) )
37	7, 36	exlimddv 1921	. . . . . 6 |- ( ph -> F e. ( X ^pm CC ) )
38		pmfun 6754	. . . . . 6 |- ( F e. ( X ^pm CC ) -> Fun F )
39	37, 38	syl 14	. . . . 5 |- ( ph -> Fun F )
40		ffvresb 5742	. . . . 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 14	. . . 4 |- ( ph -> ( ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X <-> A. x e. ( ZZ>= ` j ) ( x e. dom F /\ ( F ` x ) e. X ) ) )
42	41	rexbidv 2506	. . 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 2506	. . 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 220	. 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 167	1 |- ( ph -> E. j e. Z ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1372   E.wex 1514   e. wcel 2175   A.wral 2483   E.wrex 2484   ~Pcpw 3615   <.cop 3635   class class class wbr 4043   dom cdm 4674   ran crn 4675   |` cres 4676   Fun wfun 5264   Fn wfn 5265   -->wf 5266   ` cfv 5270  (class class class)co 5943   ^pm cpm 6735   CCcc 7922   ZZcz 9371   ZZ>=cuz 9647  TopOnctopon 14424   ~~> tclm 14601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-pm 6737  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-top 14412  df-topon 14425  df-lm 14604

This theorem is referenced by: (None)