Theorem imanonrel 44020

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

Assertion

Ref	Expression
imanonrel	⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅

Proof of Theorem imanonrel

Step	Hyp	Ref	Expression
1		df-ima 5644	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ran ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵)
2		resnonrel 44019	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅
3	2	rneqi 5892	. 2 ⊢ ran ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ran ∅
4		rn0 5881	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2763	1 ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅

Syntax hints:   = wceq 1542   ∖ cdif 3886  ∅c0 4273  ^◡ccnv 5630  ran crn 5632   ↾ cres 5633   “ cima 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by: (None)