Theorem rlimss 13407

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

Assertion

Ref	Expression
rlimss	|- ( F ~~> r A -> dom F C_ RR )

Proof of Theorem rlimss

Step	Hyp	Ref	Expression
1		rlimpm 13405	. 2 |- ( F ~~> r A -> F e. ( CC ^pm RR ) )
2		cnex 9562	. . . 4 |- CC e. _V
3		reex 9572	. . . 4 |- RR e. _V
4	2, 3	elpm2 7443	. . 3 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
5	4	simprbi 462	. 2 |- ( F e. ( CC ^pm RR ) -> dom F C_ RR )
6	1, 5	syl 16	1 |- ( F ~~> r A -> dom F C_ RR )

Syntax hints:   -> wi 4   e. wcel 1823   C_ wss 3461   class class class wbr 4439   dom cdm 4988   -->wf 5566  (class class class)co 6270   ^pm cpm 7413   CCcc 9479   RRcr 9480   ~~> r crli 13390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-pm 7415  df-rlim 13394

This theorem is referenced by:  rlimcl  13408  rlimi  13418  rlimi2  13419  rlimuni  13455  rlimres  13463  rlimeq  13474  rlimcld2  13483  rlimcn1  13493  rlimcn2  13495  rlimo1  13521  o1rlimmul  13523  rlimneg  13551  rlimsqzlem  13553  rlimno1  13558  rlimcxp  23501