Theorem imanonrel 43566

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 5636	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ran ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵)
2		resnonrel 43565	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅
3	2	rneqi 5883	. 2 ⊢ ran ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ran ∅
4		rn0 5872	. 2 ⊢ ran ∅ = ∅
5	1, 3, 4	3eqtri 2756	1 ⊢ ((𝐴 ∖ ◡◡𝐴) “ 𝐵) = ∅

Syntax hints:   = wceq 1540   ∖ cdif 3902  ∅c0 4286  ^◡ccnv 5622  ran crn 5624   ↾ cres 5625   “ cima 5626

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by: (None)