Theorem rncnv 39206

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 5877	. 2 ⊢ dom 𝐴 = ran ◡𝐴
2	1	eqcomi 2770	1 ⊢ ran ◡𝐴 = dom 𝐴

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5650  dom cdm 5651  ran crn 5652

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 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-cnv 5659  df-dm 5661  df-rn 5662

This theorem is used by:  dmcoss3  39443  symrelim  39543  symrefref2  39547