Theorem rncocnv1 5973

Description: The range of a class postcomposed with its converse is equal to its domain. (Contributed by BJ, 25-Oct-2026.)

Assertion

Ref	Expression
rncocnv1	⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴

Proof of Theorem rncocnv1

Step	Hyp	Ref	Expression
1		df-rn 5666	. . . 4 ⊢ ran 𝐴 = dom ◡𝐴
2	1	eqcomi 2772	. . 3 ⊢ dom ◡𝐴 = ran 𝐴
3		rncoeq 5971	. . 3 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴)
4	2, 3	ax-mp 5	. 2 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴
5		dfdm4 5883	. 2 ⊢ dom 𝐴 = ran ◡𝐴
6	4, 5	eqtr4i 2789	1 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴

This proof depends on syntax axioms:   = wceq 1570  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   ∘ ccom 5659

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666

This theorem is used by:  dfdm2  6299