Theorem dfdm6 34567

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

Step	Hyp	Ref	Expression
1		ecdmn0 8027	. 2 ⊢ (𝑥 ∈ dom 𝑅 ↔ [𝑥]𝑅 ≠ ∅)
2	1	abbi2i 2915	1 ⊢ dom 𝑅 = {𝑥 ∣ [𝑥]𝑅 ≠ ∅}

Syntax hints:   = wceq 1653  {cab 2785   ≠ wne 2971  ∅c0 4115  dom cdm 5312  [cec 7980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-cnv 5320  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-ec 7984

This theorem is referenced by:  dfrn6  34568