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Theorem resnonrel 43495
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 4027 . . . 4 𝐵 ⊆ V
2 ssres2 6033 . . . 4 (𝐵 ⊆ V → ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V))
31, 2ax-mp 5 . . 3 ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V)
4 cnvnonrel 43491 . . . . 5 (𝐴𝐴) = ∅
54cnveqi 5898 . . . 4 (𝐴𝐴) =
6 cnvcnv2 6223 . . . 4 (𝐴𝐴) = ((𝐴𝐴) ↾ V)
7 cnv0 6171 . . . 4 ∅ = ∅
85, 6, 73eqtr3i 2770 . . 3 ((𝐴𝐴) ↾ V) = ∅
93, 8sseqtri 4039 . 2 ((𝐴𝐴) ↾ 𝐵) ⊆ ∅
10 ss0b 4420 . 2 (((𝐴𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴𝐴) ↾ 𝐵) = ∅)
119, 10mpbi 230 1 ((𝐴𝐴) ↾ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3482  cdif 3967  wss 3970  c0 4347  ccnv 5698  cres 5701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-xp 5705  df-rel 5706  df-cnv 5707  df-res 5711
This theorem is referenced by:  imanonrel  43496
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