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Theorem resnonrel 42919
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 4001 . . . 4 𝐵 ⊆ V
2 ssres2 6003 . . . 4 (𝐵 ⊆ V → ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V))
31, 2ax-mp 5 . . 3 ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V)
4 cnvnonrel 42915 . . . . 5 (𝐴𝐴) = ∅
54cnveqi 5868 . . . 4 (𝐴𝐴) =
6 cnvcnv2 6186 . . . 4 (𝐴𝐴) = ((𝐴𝐴) ↾ V)
7 cnv0 6134 . . . 4 ∅ = ∅
85, 6, 73eqtr3i 2762 . . 3 ((𝐴𝐴) ↾ V) = ∅
93, 8sseqtri 4013 . 2 ((𝐴𝐴) ↾ 𝐵) ⊆ ∅
10 ss0b 4392 . 2 (((𝐴𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴𝐴) ↾ 𝐵) = ∅)
119, 10mpbi 229 1 ((𝐴𝐴) ↾ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3468  cdif 3940  wss 3943  c0 4317  ccnv 5668  cres 5671
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-res 5681
This theorem is referenced by:  imanonrel  42920
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