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Theorem dfdm6 38302
Description: Alternate definition of domain. (Contributed by Peter Mazsa, 2-Mar-2018.)
Assertion
Ref Expression
dfdm6 dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Distinct variable group:   𝑥,𝑅

Proof of Theorem dfdm6
StepHypRef Expression
1 ecdmn0 8794 . 2 (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅)
21eqabi 2877 1 dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  {cab 2714  wne 2940  c0 4333  dom cdm 5685  [cec 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747
This theorem is referenced by:  dfrn6  38303
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