Theorem dmcnvcnv 5958

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 6220 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 5920	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5711	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2769	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1537  ^◡ccnv 5699  dom cdm 5700  ran crn 5701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cnv 5708  df-dm 5710  df-rn 5711

This theorem is referenced by:  resdm2  6262  f1cnvcnv  6826  trrelsuperrel2dg  43633