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Theorem resnonrel 43087
Description: A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)
Assertion
Ref Expression
resnonrel ((𝐴𝐴) ↾ 𝐵) = ∅

Proof of Theorem resnonrel
StepHypRef Expression
1 ssv 3997 . . . 4 𝐵 ⊆ V
2 ssres2 6004 . . . 4 (𝐵 ⊆ V → ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V))
31, 2ax-mp 5 . . 3 ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V)
4 cnvnonrel 43083 . . . . 5 (𝐴𝐴) = ∅
54cnveqi 5871 . . . 4 (𝐴𝐴) =
6 cnvcnv2 6192 . . . 4 (𝐴𝐴) = ((𝐴𝐴) ↾ V)
7 cnv0 6140 . . . 4 ∅ = ∅
85, 6, 73eqtr3i 2761 . . 3 ((𝐴𝐴) ↾ V) = ∅
93, 8sseqtri 4009 . 2 ((𝐴𝐴) ↾ 𝐵) ⊆ ∅
10 ss0b 4393 . 2 (((𝐴𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴𝐴) ↾ 𝐵) = ∅)
119, 10mpbi 229 1 ((𝐴𝐴) ↾ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3463  cdif 3936  wss 3939  c0 4318  ccnv 5671  cres 5674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-res 5684
This theorem is referenced by:  imanonrel  43088
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