Theorem dmnonrel 43603

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 5906	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ran ◡(𝐴 ∖ ◡◡𝐴)
2		cnvnonrel 43601	. . 3 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	rneqi 5948	. 2 ⊢ ran ◡(𝐴 ∖ ◡◡𝐴) = ran ∅
4		rn0 5936	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2769	1 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅

Syntax hints:   = wceq 1540   ∖ cdif 3948  ∅c0 4333  ^◡ccnv 5684  dom cdm 5685  ran crn 5686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696

This theorem is referenced by:  rnnonrel  43604