Theorem rnnonrel 43597

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 43596	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅
2		dm0rn0 5942	. 2 ⊢ (dom (𝐴 ∖ ◡◡𝐴) = ∅ ↔ ran (𝐴 ∖ ◡◡𝐴) = ∅)
3	1, 2	mpbi 230	1 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1539   ∖ cdif 3963  ∅c0 4342  ^◡ccnv 5692  dom cdm 5693  ran crn 5694

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152  df-opab 5214  df-xp 5699  df-rel 5700  df-cnv 5701  df-dm 5703  df-rn 5704

This theorem is referenced by:  fvnonrel  43603