Theorem dmnonrel 41087

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 5793	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ran ◡(𝐴 ∖ ◡◡𝐴)
2		cnvnonrel 41085	. . 3 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	rneqi 5835	. 2 ⊢ ran ◡(𝐴 ∖ ◡◡𝐴) = ran ∅
4		rn0 5824	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2770	1 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1539   ∖ cdif 3880  ∅c0 4253  ^◡ccnv 5579  dom cdm 5580  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  rnnonrel  41088