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Theorem dfrn6 37768
Description: Alternate definition of range. (Contributed by Peter Mazsa, 1-Aug-2018.)
Assertion
Ref Expression
dfrn6 ran 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Distinct variable group:   𝑥,𝑅

Proof of Theorem dfrn6
StepHypRef Expression
1 df-rn 5683 . 2 ran 𝑅 = dom 𝑅
2 dfdm6 37767 . 2 dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
31, 2eqtri 2756 1 ran 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  {cab 2705  wne 2936  c0 4318  ccnv 5671  dom cdm 5672  ran crn 5673  [cec 8716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8720
This theorem is referenced by:  rnxrn  37864
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