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Theorem elrn 5907
Description: Membership in a range. (Contributed by NM, 2-Apr-2004.)
Hypothesis
Ref Expression
elrn.1 𝐴 ∈ V
Assertion
Ref Expression
elrn (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elrn
StepHypRef Expression
1 elrn.1 . 2 𝐴 ∈ V
2 elrng 5905 . 2 (𝐴 ∈ V → (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴))
31, 2ax-mp 5 1 (𝐴 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wex 1776  wcel 2106  Vcvv 3478   class class class wbr 5148  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  dmcosseq  5990  dmcosseqOLD  5991  inisegn0  6119  rnco  6274  dffo4  7123  fvclss  7261  rntpos  8263  fpwwe2lem10  10678  fpwwe2lem11  10679  fclim  15586  perfdvf  25953  dftr6  35731  dffr5  35734  brsset  35871  dfon3  35874  brtxpsd  35876  dffix2  35887  elsingles  35900  dfrdg4  35933  undmrnresiss  43594
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