Theorem errn 6519

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 4614	. 2 |- ran R = dom `' R
2		ercnv 6518	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4805	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6507	. . 3 |- ( R Er A -> dom R = A )
5	3, 4	eqtrd 2198	. 2 |- ( R Er A -> dom `' R = A )
6	1, 5	syl5eq 2210	1 |- ( R Er A -> ran R = A )

Syntax hints:   -> wi 4   = wceq 1343   `'ccnv 4602   dom cdm 4603   ran crn 4604   Er wer 6494

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4099  ax-pow 4152  ax-pr 4186

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-v 2727  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-br 3982  df-opab 4043  df-xp 4609  df-rel 4610  df-cnv 4611  df-dm 4613  df-rn 4614  df-er 6497

This theorem is referenced by:  erssxp  6520  ecss  6538  uniqs2  6557