Theorem errn 6611

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 4671	. 2 ⊢ ran 𝑅 = dom ◡𝑅
2		ercnv 6610	. . . 4 ⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅)
3	2	dmeqd 4865	. . 3 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = dom 𝑅)
4		erdm 6599	. . 3 ⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴)
5	3, 4	eqtrd 2226	. 2 ⊢ (𝑅 Er 𝐴 → dom ◡𝑅 = 𝐴)
6	1, 5	eqtrid 2238	1 ⊢ (𝑅 Er 𝐴 → ran 𝑅 = 𝐴)

Syntax hints:   → wi 4   = wceq 1364  ^◡ccnv 4659  dom cdm 4660  ran crn 4661   Er wer 6586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671  df-er 6589

This theorem is referenced by:  erssxp  6612  ecss  6632  uniqs2  6651