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Theorem dfdm6 34414
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 7959 . 2 (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅)
21abbi2i 2877 1 dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  {cab 2747  wne 2933  c0 4059  dom cdm 5267  [cec 7912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-cnv 5275  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-ec 7916
This theorem is referenced by:  dfrn6  34415
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