Theorem errn 6829

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 4785	. 2 |- ran R = dom `' R
2		ercnv 6828	. . . 4 |- ( R Er A -> `' R = R )
3	2	dmeqd 4983	. . 3 |- ( R Er A -> dom `' R = dom R )
4		erdm 6817	. . 3 |- ( R Er A -> dom R = A )
5	3, 4	eqtrd 2271	. 2 |- ( R Er A -> dom `' R = A )
6	1, 5	eqtrid 2283	1 |- ( R Er A -> ran R = A )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   `'ccnv 4773   dom cdm 4774   ran crn 4775   Er wer 6804

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785  df-er 6807

This theorem is used by:  erssxp  6830  ecss  6850  uniqs2  6869