Theorem lmrcl 21035

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 21033	. . 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 5631	. 2 |- dom ~~> t C_ Top
3		df-br 4654	. . 3 |- ( F ( ~~> t ` J ) P <-> <. F , P >. e. ( ~~> t ` J ) )
4		elfvdm 6220	. . 3 |- ( <. F , P >. e. ( ~~> t ` J ) -> J e. dom ~~> t )
5	3, 4	sylbi 207	. 2 |- ( F ( ~~> t ` J ) P -> J e. dom ~~> t )
6	2, 5	sseldi 3601	1 |- ( F ( ~~> t ` J ) P -> J e. Top )

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   A.wral 2912   E.wrex 2913   <.cop 4183   U.cuni 4436   class class class wbr 4653   {copab 4712   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   ZZ>=cuz 11687   Topctop 20698   ~~> 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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896  df-lm 21033

This theorem is referenced by:  lmcvg  21066