Theorem errn 6700

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 4729	. 2 |- ran R = dom `' R
2		ercnv 6699	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4924	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6688	. . 3 |- ( R Er A -> dom R = A )
5	3, 4	eqtrd 2262	. 2 |- ( R Er A -> dom `' R = A )
6	1, 5	eqtrid 2274	1 |- ( R Er A -> ran R = A )

Syntax hints:   -> wi 4   = wceq 1395   `'ccnv 4717   dom cdm 4718   ran crn 4719   Er wer 6675

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-dm 4728  df-rn 4729  df-er 6678

This theorem is referenced by:  erssxp  6701  ecss  6721  uniqs2  6740