Theorem lmrcl 15293

Description: Reverse closure for the convergence relation. (Contributed by Mario Carneiro, 7-Sep-2015.)

Assertion

Ref	Expression
lmrcl	|- ( F ( ~~> t ` J ) P -> J e. Top )

Proof of Theorem lmrcl

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

Step	Hyp	Ref	Expression
1		df-lm 15291	. . 3 |- ~~> t = ( j e. Top |-> { <. f , x >. | ( f e. ( U. j ^pm CC ) /\ x e. U. j /\ A. u e. j ( x e. u -> E. y e. ran ZZ>= ( f |` y ) : y --> u ) ) } )
2	1	dmmptss 5284	. 2 |- dom ~~> t C_ Top
3		df-br 4131	. . 3 |- ( F ( ~~> t ` J ) P <-> <. F , P >. e. ( ~~> t ` J ) )
4	1	funmpt2 5416	. . . . 5 |- Fun ~~> t
5		funrel 5394	. . . . 5 |- ( Fun ~~> t -> Rel ~~> t )
6	4, 5	ax-mp 5	. . . 4 |- Rel ~~> t
7		relelfvdm 5727	. . . 4 |- ( ( Rel ~~> t /\ <. F , P >. e. ( ~~> t ` J ) ) -> J e. dom ~~> t )
8	6, 7	mpan 428	. . 3 |- ( <. F , P >. e. ( ~~> t ` J ) -> J e. dom ~~> t )
9	3, 8	sylbi 121	. 2 |- ( F ( ~~> t ` J ) P -> J e. dom ~~> t )
10	2, 9	sselid 3246	1 |- ( F ( ~~> t ` J ) P -> J e. Top )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   e. wcel 2209   A.wral 2528   E.wrex 2529   <.cop 3712   U.cuni 3935   class class class wbr 4130   {copab 4191   dom cdm 4774   ran crn 4775   |` cres 4776   Rel wrel 4779   Fun wfun 5371   -->wf 5373   ` cfv 5377  (class class class)co 6085   ^pm cpm 6923   CCcc 8177   ZZ>=cuz 9921   Topctop 15098   ~~> tclm 15288

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fv 5385  df-lm 15291

This theorem is used by:  lmcvg  15318  lmtopcnp  15351