Theorem rncnvcnv 4810

Description: The range of the double converse of a class. (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
rncnvcnv	|- ran `' `' A = ran A

Proof of Theorem rncnvcnv

Step	Hyp	Ref	Expression
1		df-rn 4596	. 2 |- ran A = dom `' A
2		dfdm4 4777	. 2 |- dom `' A = ran `' `' A
3	1, 2	eqtr2i 2179	1 |- ran `' `' A = ran A

Syntax hints:   = wceq 1335   `'ccnv 4584   dom cdm 4585   ran crn 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4593  df-dm 4595  df-rn 4596

This theorem is referenced by:  rnresv  5044