Theorem errn 6802

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 4765	. 2 |- ran R = dom `' R
2		ercnv 6801	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4963	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6790	. . 3 |- ( R Er A -> dom R = A )
5	3, 4	eqtrd 2267	. 2 |- ( R Er A -> dom `' R = A )
6	1, 5	eqtrid 2279	1 |- ( R Er A -> ran R = A )

Syntax hints:   -> wi 4   = wceq 1398   `'ccnv 4753   dom cdm 4754   ran crn 4755   Er wer 6777

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-dm 4764  df-rn 4765  df-er 6780

This theorem is referenced by:  erssxp  6803  ecss  6823  uniqs2  6842