Theorem dmcnvcnv 4803

Description: The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5029). (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
dmcnvcnv	|- dom `' `' A = dom A

Proof of Theorem dmcnvcnv

Step	Hyp	Ref	Expression
1		dfdm4 4771	. 2 |- dom A = ran `' A
2		df-rn 4590	. 2 |- ran `' A = dom `' `' A
3	1, 2	eqtr2i 2176	1 |- dom `' `' A = dom A

Syntax hints:   = wceq 1332   `'ccnv 4578   dom cdm 4579   ran crn 4580

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-br 3962  df-opab 4022  df-cnv 4587  df-dm 4589  df-rn 4590

This theorem is referenced by:  resdm2  5069  f1cnvcnv  5379