Theorem dmnonrel 44034

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 5837	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ran ◡(𝐴 ∖ ◡◡𝐴)
2		cnvnonrel 44032	. . 3 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	rneqi 5879	. 2 ⊢ ran ◡(𝐴 ∖ ◡◡𝐴) = ran ∅
4		rn0 5868	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2766	1 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1547   ∖ cdif 3880  ∅c0 4261  ^◡ccnv 5617  dom cdm 5618  ran crn 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629

This theorem is referenced by:  rnnonrel  44035