Theorem rlimpm 14231

Description: Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)

Assertion

Ref	Expression
rlimpm	|- ( F ~~> r A -> F e. ( CC ^pm RR ) )

Proof of Theorem rlimpm

Dummy variables w f x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rlim 14220	. . . . 5 |- ~~> r = { <. f , x >. | ( ( f e. ( CC ^pm RR ) /\ x e. CC ) /\ A. y e. RR+ E. z e. RR A. w e. dom f ( z <_ w -> ( abs ` ( ( f ` w ) - x ) ) < y ) ) }
2		opabssxp 5193	. . . . 5 |- { <. f , x >. | ( ( f e. ( CC ^pm RR ) /\ x e. CC ) /\ A. y e. RR+ E. z e. RR A. w e. dom f ( z <_ w -> ( abs ` ( ( f ` w ) - x ) ) < y ) ) } C_ ( ( CC ^pm RR ) X. CC )
3	1, 2	eqsstri 3635	. . . 4 |- ~~> r C_ ( ( CC ^pm RR ) X. CC )
4		dmss 5323	. . . 4 |- ( ~~> r C_ ( ( CC ^pm RR ) X. CC ) -> dom ~~> r C_ dom ( ( CC ^pm RR ) X. CC ) )
5	3, 4	ax-mp 5	. . 3 |- dom ~~> r C_ dom ( ( CC ^pm RR ) X. CC )
6		dmxpss 5565	. . 3 |- dom ( ( CC ^pm RR ) X. CC ) C_ ( CC ^pm RR )
7	5, 6	sstri 3612	. 2 |- dom ~~> r C_ ( CC ^pm RR )
8		rlimrel 14224	. . 3 |- Rel ~~> r
9	8	releldmi 5362	. 2 |- ( F ~~> r A -> F e. dom ~~> r )
10	7, 9	sseldi 3601	1 |- ( F ~~> r A -> F e. ( CC ^pm RR ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   class class class wbr 4653   {copab 4712   X. cxp 5112   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832   abscabs 13974   ~~> r crli 14216

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rlim 14220

This theorem is referenced by:  rlimf  14232  rlimss  14233  rlimclim1  14276