Theorem dmcnvcnv 5939

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 6198 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 5902	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5693	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2757	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1533  ^◡ccnv 5681  dom cdm 5682  ran crn 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-cnv 5690  df-dm 5692  df-rn 5693

This theorem is referenced by:  resdm2  6240  f1cnvcnv  6808  trrelsuperrel2dg  43132