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Theorem dfrn6 38304
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 5695 . 2 ran 𝑅 = dom 𝑅
2 dfdm6 38303 . 2 dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
31, 2eqtri 2764 1 ran 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  {cab 2713  wne 2939  c0 4332  ccnv 5683  dom cdm 5684  ran crn 5685  [cec 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-cnv 5692  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ec 8748
This theorem is referenced by:  rnxrn  38400
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