Theorem rnnonrel 44300

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 44299	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅
2		dm0rn0 5916	. 2 ⊢ (dom (𝐴 ∖ ◡◡𝐴) = ∅ ↔ ran (𝐴 ∖ ◡◡𝐴) = ∅)
3	1, 2	mpbi 233	1 ⊢ ran (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1570   ∖ cdif 3903  ∅c0 4287  ^◡ccnv 5662  dom cdm 5663  ran crn 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675

This theorem is referenced by:  fvnonrel  44306