Theorem rncnvcnv 5930

Description: The range of the double converse of a class is equal to its range (even when that class in not a relation). (Contributed by NM, 8-Apr-2007.)

Assertion

Ref	Expression
rncnvcnv	⊢ ran ◡◡𝐴 = ran 𝐴

Proof of Theorem rncnvcnv

Step	Hyp	Ref	Expression
1		df-rn 5683	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		dfdm4 5892	. 2 ⊢ dom ◡𝐴 = ran ◡◡𝐴
3	1, 2	eqtr2i 2755	1 ⊢ ran ◡◡𝐴 = ran 𝐴

Syntax hints:   = wceq 1534  ^◡ccnv 5671  dom cdm 5672  ran crn 5673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5144  df-opab 5206  df-cnv 5680  df-dm 5682  df-rn 5683

This theorem is referenced by:  rnresv  6202  trrelsuperrel2dg  43372