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

Step	Hyp	Ref	Expression
1		df-rn 5695	. 2 ⊢ ran 𝑅 = dom ◡𝑅
2		dfdm6 38303	. 2 ⊢ dom ◡𝑅 = {𝑥 ∣ [𝑥]◡𝑅 ≠ ∅}
3	1, 2	eqtri 2764	1 ⊢ ran 𝑅 = {𝑥 ∣ [𝑥]◡𝑅 ≠ ∅}

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