Theorem dmcnvcnv 5888

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 6153 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 5850	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5642	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2760	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1542  ^◡ccnv 5630  dom cdm 5631  ran crn 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-cnv 5639  df-dm 5641  df-rn 5642

This theorem is referenced by:  resdm2  6195  f1cnvcnv  6745  trrelsuperrel2dg  44098