Theorem dmcnvcnv 4901

Description: The domain of the double converse of a class (which doesn't have to be a relation as in dfrel2 5132). (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
dmcnvcnv	⊢ dom ◡◡𝐴 = dom 𝐴

Proof of Theorem dmcnvcnv

Step	Hyp	Ref	Expression
1		dfdm4 4869	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 4685	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2226	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1372  ^◡ccnv 4673  dom cdm 4674  ran crn 4675

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-cnv 4682  df-dm 4684  df-rn 4685

This theorem is referenced by:  resdm2  5172  f1cnvcnv  5491