Theorem errn 6557

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 4638	. 2 |- ran R = dom `' R
2		ercnv 6556	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4830	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6545	. . 3 |- ( R Er A -> dom R = A )
5	3, 4	eqtrd 2210	. 2 |- ( R Er A -> dom `' R = A )
6	1, 5	eqtrid 2222	1 |- ( R Er A -> ran R = A )

Syntax hints:   -> wi 4   = wceq 1353   `'ccnv 4626   dom cdm 4627   ran crn 4628   Er wer 6532

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-dm 4637  df-rn 4638  df-er 6535

This theorem is referenced by:  erssxp  6558  ecss  6576  uniqs2  6595