Theorem lmrcl 14748

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 14747	. . 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 5193	. 2 |- dom ~~> t C_ Top
3		df-br 4055	. . 3 |- ( F ( ~~> t ` J ) P <-> <. F , P >. e. ( ~~> t ` J ) )
4	1	funmpt2 5324	. . . . 5 |- Fun ~~> t
5		funrel 5302	. . . . 5 |- ( Fun ~~> t -> Rel ~~> t )
6	4, 5	ax-mp 5	. . . 4 |- Rel ~~> t
7		relelfvdm 5626	. . . 4 |- ( ( Rel ~~> t /\ <. F , P >. e. ( ~~> t ` J ) ) -> J e. dom ~~> t )
8	6, 7	mpan 424	. . 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 3195	1 |- ( F ( ~~> t ` J ) P -> J e. Top )

Syntax hints:   -> wi 4   /\ w3a 981   e. wcel 2177   A.wral 2485   E.wrex 2486   <.cop 3641   U.cuni 3859   class class class wbr 4054   {copab 4115   dom cdm 4688   ran crn 4689   |` cres 4690   Rel wrel 4693   Fun wfun 5279   -->wf 5281   ` cfv 5285  (class class class)co 5962   ^pm cpm 6754   CCcc 7953   ZZ>=cuz 9678   Topctop 14554   ~~> tclm 14744

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fv 5293  df-lm 14747

This theorem is referenced by:  lmcvg  14774  lmtopcnp  14807