Theorem errn 6767

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 4742	. 2 |- ran R = dom `' R
2		ercnv 6766	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4939	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6755	. . 3 |- ( R Er A -> dom R = A )
5	3, 4	eqtrd 2264	. 2 |- ( R Er A -> dom `' R = A )
6	1, 5	eqtrid 2276	1 |- ( R Er A -> ran R = A )

Syntax hints:   -> wi 4   = wceq 1398   `'ccnv 4730   dom cdm 4731   ran crn 4732   Er wer 6742

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742  df-er 6745

This theorem is referenced by:  erssxp  6768  ecss  6788  uniqs2  6807