Theorem rnnonrel 44127

Description: The range of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)

Assertion

Ref	Expression
rnnonrel	⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅

Proof of Theorem rnnonrel

Step	Hyp	Ref	Expression
1		dmnonrel 44126	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅
2		dm0rn0 5896	. 2 ⊢ (dom (𝐴 ∖ ◡◡𝐴) = ∅ ↔ ran (𝐴 ∖ ◡◡𝐴) = ∅)
3	1, 2	mpbi 232	1 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1559   ∖ cdif 3899  ∅c0 4283  ^◡ccnv 5642  dom cdm 5643  ran crn 5644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-rel 5650  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655

This theorem is referenced by:  fvnonrel  44133