Theorem lmff 15240

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 4957	. . . . . . 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 4115	. . . . . 6 |- ( F ( ~~> t ` J ) y <-> <. F , y >. e. ( ~~> t ` J ) )
6	5	exbii 1654	. . . . 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 15236	. . . . . 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 2298	. . . . . . 7 |- ( j = X -> ( y e. j <-> y e. X ) )
12		feq3 5498	. . . . . . . 8 |- ( j = X -> ( ( F |` x ) : x --> j <-> ( F |` x ) : x --> X ) )
13	12	rexbidv 2545	. . . . . . 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 15204	. . . . . . . 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 1038	. . . . . 6 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> A. j e. J ( y e. j -> E. x e. ran ZZ>= ( F |` x ) : x --> j ) )
18		toponmax 15016	. . . . . . . 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 2928	. . . . 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 1950	. . 3 |- ( ph -> E. x e. ran ZZ>= ( F |` x ) : x --> X )
24		uzf 9874	. . . 4 |- ZZ>= : ZZ --> ~P ZZ
25		ffn 5513	. . . 4 |- ( ZZ>= : ZZ --> ~P ZZ -> ZZ>= Fn ZZ )
26		reseq2 5038	. . . . . 6 |- ( x = ( ZZ>= ` j ) -> ( F |` x ) = ( F |` ( ZZ>= ` j ) ) )
27		id 19	. . . . . 6 |- ( x = ( ZZ>= ` j ) -> x = ( ZZ>= ` j ) )
28	26, 27	feq12d 5503	. . . . 5 |- ( x = ( ZZ>= ` j ) -> ( ( F |` x ) : x --> X <-> ( F |` ( ZZ>= ` j ) ) : ( ZZ>= ` j ) --> X ) )
29	28	rexrn 5819	. . . 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 11700	. . . 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 1036	. . . . . . 7 |- ( ( ph /\ F ( ~~> t ` J ) y ) -> F e. ( X ^pm CC ) )
37	7, 36	exlimddv 1950	. . . . . 6 |- ( ph -> F e. ( X ^pm CC ) )
38		pmfun 6915	. . . . . 6 |- ( F e. ( X ^pm CC ) -> Fun F )
39	37, 38	syl 14	. . . . 5 |- ( ph -> Fun F )
40		ffvresb 5845	. . . . 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 2545	. . 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 2545	. . 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 1005   = wceq 1398   E.wex 1541   e. wcel 2205   A.wral 2522   E.wrex 2523   ~Pcpw 3674   <.cop 3697   class class class wbr 4114   dom cdm 4754   ran crn 4755   |` cres 4756   Fun wfun 5351   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058   ^pm cpm 6896   CCcc 8141   ZZcz 9594   ZZ>=cuz 9871  TopOnctopon 15001   ~~> tclm 15178

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pm 6898  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-top 14989  df-topon 15002  df-lm 15181

This theorem is referenced by: (None)