Theorem rnnonrel 43542

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 43541	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅
2		dm0rn0 5915	. 2 ⊢ (dom (𝐴 ∖ ◡◡𝐴) = ∅ ↔ ran (𝐴 ∖ ◡◡𝐴) = ∅)
3	1, 2	mpbi 230	1 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1539   ∖ cdif 3928  ∅c0 4313  ^◡ccnv 5664  dom cdm 5665  ran crn 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676

This theorem is referenced by:  fvnonrel  43548