Theorem dmcnvcnv 4887

Description: The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5117). (Contributed by NM, 8-Apr-2007.)

Assertion

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

Proof of Theorem dmcnvcnv

Step	Hyp	Ref	Expression
1		dfdm4 4855	. 2 |- dom A = ran `' A
2		df-rn 4671	. 2 |- ran `' A = dom `' `' A
3	1, 2	eqtr2i 2215	1 |- dom `' `' A = dom A

Syntax hints:   = wceq 1364   `'ccnv 4659   dom cdm 4660   ran crn 4661

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by:  resdm2  5157  f1cnvcnv  5471