Theorem errn 8478

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 5591	. 2 ⊢ ran 𝑅 = dom ◡𝑅
2		ercnv 8477	. . . 4 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)
3	2	dmeqd 5803	. . 3 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = dom 𝑅)
4		erdm 8466	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5	3, 4	eqtrd 2778	. 2 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = 𝐴)
6	1, 5	eqtrid 2790	1 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1539  ^◡ccnv 5579  dom cdm 5580  ran crn 5581   Er wer 8453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-er 8456

This theorem is referenced by:  erssxp  8479  ecss  8502  uniqs2  8526  sylow2a  19139