Theorem dmcnvcnv 4852

Description: The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5080). (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
dmcnvcnv	⊢ dom ◡◡𝐴 = dom 𝐴

Proof of Theorem dmcnvcnv

Step	Hyp	Ref	Expression
1		dfdm4 4820	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 4638	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2199	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1353  ^◡ccnv 4626  dom cdm 4627  ran crn 4628

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-cnv 4635  df-dm 4637  df-rn 4638

This theorem is referenced by:  resdm2  5120  f1cnvcnv  5433