Theorem errn 8765

Description: The range and domain of an equivalence relation are equal. (Contributed by Rodolfo Medina, 11-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)

Assertion

Ref	Expression
errn	⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)

Proof of Theorem errn

Step	Hyp	Ref	Expression
1		df-rn 5699	. 2 ⊢ ran 𝑅 = dom ◡𝑅
2		ercnv 8764	. . . 4 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)
3	2	dmeqd 5918	. . 3 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = dom 𝑅)
4		erdm 8753	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5	3, 4	eqtrd 2774	. 2 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = 𝐴)
6	1, 5	eqtrid 2786	1 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1536  ^◡ccnv 5687  dom cdm 5688  ran crn 5689   Er wer 8740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-er 8743

This theorem is referenced by:  erssxp  8766  ecss  8791  uniqs2  8817  sylow2a  19651