Theorem dmcnvcnv 5767

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 6013 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 5728	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2		df-rn 5530	. 2 ⊢ ran ◡𝐴 = dom ◡◡𝐴
3	1, 2	eqtr2i 2822	1 ⊢ dom ◡◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1538  ^◡ccnv 5518  dom cdm 5519  ran crn 5520

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-cnv 5527  df-dm 5529  df-rn 5530

This theorem is referenced by:  resdm2  6055  f1cnvcnv  6559  trrelsuperrel2dg  40372