Theorem rncnvcnv 5928

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 5676	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		dfdm4 5889	. 2 ⊢ dom ◡𝐴 = ran ◡◡𝐴
3	1, 2	eqtr2i 2794	1 ⊢ ran ◡◡𝐴 = ran 𝐴

Syntax hints:   = wceq 1568  ^◡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 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-cnv 5673  df-dm 5675  df-rn 5676

This theorem is referenced by:  rnresv  6204  trrelsuperrel2dg  44349