Theorem dfrn6 38559

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 5643	. 2 ⊢ ran 𝑅 = dom ◡𝑅
2		dfdm6 38558	. 2 ⊢ dom ◡𝑅 = {𝑥 ∣ [𝑥]◡𝑅 ≠ ∅}
3	1, 2	eqtri 2760	1 ⊢ ran 𝑅 = {𝑥 ∣ [𝑥]◡𝑅 ≠ ∅}

Syntax hints:   = wceq 1542  {cab 2715   ≠ wne 2933  ∅c0 4287  ^◡ccnv 5631  dom cdm 5632  ran crn 5633  [cec 8643

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647

This theorem is referenced by:  rnxrn  38672