Theorem lmff 13416

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 4819	. . . . . . 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 4001	. . . . . 6 |- ( F ( ~~> t ` J ) y <-> <. F , y >. e. ( ~~> t ` J ) )
6	5	exbii 1605	. . . . 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 13412	. . . . . 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 2241	. . . . . . 7 |- ( j = X -> ( y e. j <-> y e. X ) )
12		feq3 5346	. . . . . . . 8 |- ( j = X -> ( ( F |` x ) : x --> j <-> ( F |` x ) : x --> X ) )
13	12	rexbidv 2478	. . . . . . 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 13380	. . . . . . . 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 1011	. . . . . 6 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> A. j e. J ( y e. j -> E. x e. ran ZZ>= ( F |` x ) : x --> j ) )
18		toponmax 13190	. . . . . . . 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 2846	. . . . 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 1898	. . 3 |- ( ph -> E. x e. ran ZZ>= ( F |` x ) : x --> X )
24		uzf 9520	. . . 4 |- ZZ>= : ZZ --> ~P ZZ
25		ffn 5361	. . . 4 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
26		reseq2 4898	. . . . . 6 |- ( x = ( ZZ>= ` j ) -> ( F |` x ) = ( F |` ( ZZ>= ` j ) ) )
27		id 19	. . . . . 6 |- ( x = ( ZZ>= ` j ) -> x = ( ZZ>= ` j ) )
28	26, 27	feq12d 5351	. . . . 5 |- ( x = ( ZZ>= ` j ) -> ( ( F |` x ) : x --> X <-> ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) )
29	28	rexrn 5649	. . . 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 10983	. . . 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 1009	. . . . . . 7 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> F e. ( X ^pm CC ) )
37	7, 36	exlimddv 1898	. . . . . 6 |- ( ph -> F e. ( X ^pm CC ) )
38		pmfun 6662	. . . . . 6 |- ( F e. ( X ^pm CC ) -> Fun F )
39	37, 38	syl 14	. . . . 5 |- ( ph -> Fun F )
40		ffvresb 5675	. . . . 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 2478	. . 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 2478	. . 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 978   = wceq 1353   E.wex 1492   e. wcel 2148   A.wral 2455   E.wrex 2456   ~Pcpw 3574   <.cop 3594   class class class wbr 4000   dom cdm 4623   ran crn 4624   |` cres 4625   Fun wfun 5206   Fn wfn 5207   -->wf 5208   ` cfv 5212  (class class class)co 5869   ^pm cpm 6643   CCcc 7800   ZZcz 9242   ZZ>=cuz 9517  TopOnctopon 13175   ~~> tclm 13354

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-pm 6645  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-top 13163  df-topon 13176  df-lm 13357

This theorem is referenced by: (None)