Theorem errn 6723

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 4736	. 2 |- ran R = dom `' R
2		ercnv 6722	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4933	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6711	. . 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 1397   `'ccnv 4724   dom cdm 4725   ran crn 4726   Er wer 6698

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-er 6701

This theorem is referenced by:  erssxp  6724  ecss  6744  uniqs2  6763