Theorem rnnonrel 44036

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 44035	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅
2		dm0rn0 5873	. 2 ⊢ (dom (𝐴 ∖ ◡◡𝐴) = ∅ ↔ ran (𝐴 ∖ ◡◡𝐴) = ∅)
3	1, 2	mpbi 230	1 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1542   ∖ cdif 3887  ∅c0 4274  ^◡ccnv 5623  dom cdm 5624  ran crn 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635

This theorem is referenced by:  fvnonrel  44042