Theorem rncnvcnv 5882

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 5634	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		dfdm4 5843	. 2 ⊢ dom ◡𝐴 = ran ◡◡𝐴
3	1, 2	eqtr2i 2759	1 ⊢ ran ◡◡𝐴 = ran 𝐴

Syntax hints:   = wceq 1542  ^◡ccnv 5622  dom cdm 5623  ran crn 5624

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 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

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 2714  df-cleq 2727  df-clel 2810  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-cnv 5631  df-dm 5633  df-rn 5634

This theorem is referenced by:  rnresv  6158  trrelsuperrel2dg  43949