Theorem dmnonrel 44349

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 5887	. 2 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ran ◡(𝐴 ∖ ◡◡𝐴)
2		cnvnonrel 44347	. . 3 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
3	2	rneqi 5929	. 2 ⊢ ran ◡(𝐴 ∖ ◡◡𝐴) = ran ∅
4		rn0 5918	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2792	1 ⊢ dom (𝐴 ∖ ◡◡𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3903  ∅c0 4286  ^◡ccnv 5662  dom cdm 5663  ran crn 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675

This theorem is used by:  rnnonrel  44350