Theorem dmnonrel 43548

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

Assertion

Ref	Expression
dmnonrel	⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅

Proof of Theorem dmnonrel

Step	Hyp	Ref	Expression
1		dfdm4 5915	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ran ◡(𝐴 ∖ ◡◡𝐴)
2		cnvnonrel 43546	. . 3 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	rneqi 5957	. 2 ⊢ ran ◡(𝐴 ∖ ◡◡𝐴) = ran ∅
4		rn0 5945	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2772	1 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1537   ∖ cdif 3973  ∅c0 4352  ^◡ccnv 5694  dom cdm 5695  ran crn 5696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5701  df-rel 5702  df-cnv 5703  df-dm 5705  df-rn 5706

This theorem is referenced by:  rnnonrel  43549