Theorem errn 6632

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 4684	. 2 |- ran R = dom `' R
2		ercnv 6631	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4878	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6620	. . 3 |- ( R Er A -> dom R = A )
5	3, 4	eqtrd 2237	. 2 |- ( R Er A -> dom `' R = A )
6	1, 5	eqtrid 2249	1 |- ( R Er A -> ran R = A )

Syntax hints:   -> wi 4   = wceq 1372   `'ccnv 4672   dom cdm 4673   ran crn 4674   Er wer 6607

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4679  df-rel 4680  df-cnv 4681  df-dm 4683  df-rn 4684  df-er 6610

This theorem is referenced by:  erssxp  6633  ecss  6653  uniqs2  6672