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Theorem resnonrel 42945
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 4002 . . . 4 𝐵 ⊆ V
2 ssres2 6007 . . . 4 (𝐵 ⊆ V → ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V))
31, 2ax-mp 5 . . 3 ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V)
4 cnvnonrel 42941 . . . . 5 (𝐴𝐴) = ∅
54cnveqi 5871 . . . 4 (𝐴𝐴) =
6 cnvcnv2 6191 . . . 4 (𝐴𝐴) = ((𝐴𝐴) ↾ V)
7 cnv0 6139 . . . 4 ∅ = ∅
85, 6, 73eqtr3i 2763 . . 3 ((𝐴𝐴) ↾ V) = ∅
93, 8sseqtri 4014 . 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 1534  Vcvv 3469  cdif 3941  wss 3944  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 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-res 5684
This theorem is referenced by:  imanonrel  42946
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