Theorem rncnv 38996

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 5890	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2	1	eqcomi 2775	1 ⊢ ran ◡𝐴 = dom 𝐴

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5665  dom cdm 5666  ran crn 5667

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-cnv 5674  df-dm 5676  df-rn 5677

This theorem is used by:  dmcoss3  39233  symrelim  39333  symrefref2  39337