Theorem rlimss 13097

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 13095	. 2 |- ( F ~~> r A -> F e. ( CC ^pm RR ) )
2		cnex 9473	. . . 4 |- CC e. _V
3		reex 9483	. . . 4 |- RR e. _V
4	2, 3	elpm2 7353	. . 3 |- ( F e. ( CC ^pm RR ) <-> ( F : dom F --> CC /\ dom F C_ RR ) )
5	4	simprbi 464	. 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 1758   C_ wss 3435   class class class wbr 4399   dom cdm 4947   -->wf 5521  (class class class)co 6199   ^pm cpm 7324   CCcc 9390   RRcr 9391   ~~> r crli 13080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-pm 7326  df-rlim 13084

This theorem is referenced by:  rlimcl  13098  rlimi  13108  rlimi2  13109  rlimuni  13145  rlimres  13153  rlimeq  13164  rlimcld2  13173  rlimcn1  13183  rlimcn2  13185  rlimo1  13211  o1rlimmul  13213  rlimneg  13241  rlimsqzlem  13243  rlimno1  13248  rlimcxp  22499