Theorem errn 6459

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 4558	. 2 |- ran R = dom `' R
2		ercnv 6458	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4749	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6447	. . 3 |- ( R Er A -> dom R = A )
5	3, 4	eqtrd 2173	. 2 |- ( R Er A -> dom `' R = A )
6	1, 5	syl5eq 2185	1 |- ( R Er A -> ran R = A )

Syntax hints:   -> wi 4   = wceq 1332   `'ccnv 4546   dom cdm 4547   ran crn 4548   Er wer 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-er 6437

This theorem is referenced by:  erssxp  6460  ecss  6478  uniqs2  6497