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Theorem dfrn6 35719
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 5534 . 2 ran 𝑅 = dom 𝑅
2 dfdm6 35718 . 2 dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
31, 2eqtri 2824 1 ran 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  {cab 2779  wne 2990  c0 4246  ccnv 5522  dom cdm 5523  ran crn 5524  [cec 8274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ec 8278
This theorem is referenced by:  rnxrn  35805
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