Theorem dmcnvcnv 5924

Description: The domain of the double converse of a class is equal to its domain (even when that class in not a relation, in which case dfrel2 6189 gives another proof). (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
dmcnvcnv	⊢ dom ◡◡𝐴 = dom 𝐴

Proof of Theorem dmcnvcnv

Step	Hyp	Ref	Expression
1		dfdm4 5886	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5676	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2758	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1539  ^◡ccnv 5664  dom cdm 5665  ran crn 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-cnv 5673  df-dm 5675  df-rn 5676

This theorem is referenced by:  resdm2  6231  f1cnvcnv  6793  trrelsuperrel2dg  43661