Theorem rncnv 35035

Description: Range of converse is the domain. (Contributed by Peter Mazsa, 12-Feb-2018.)

Assertion

Ref	Expression
rncnv	⊢ ran ◡𝐴 = dom 𝐴

Proof of Theorem rncnv

Step	Hyp	Ref	Expression
1		dfdm4 5611	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2	1	eqcomi 2782	1 ⊢ ran ◡𝐴 = dom 𝐴

Syntax hints:   = wceq 1508  ^◡ccnv 5403  dom cdm 5404  ran crn 5405

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pr 5183

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-br 4927  df-opab 4989  df-cnv 5412  df-dm 5414  df-rn 5415

This theorem is referenced by:  dmcoss3  35171  symrelim  35273  symrefref2  35277