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Theorem resnonrel 43556
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 4033 . . . 4 𝐵 ⊆ V
2 ssres2 6036 . . . 4 (𝐵 ⊆ V → ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V))
31, 2ax-mp 5 . . 3 ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V)
4 cnvnonrel 43552 . . . . 5 (𝐴𝐴) = ∅
54cnveqi 5899 . . . 4 (𝐴𝐴) =
6 cnvcnv2 6226 . . . 4 (𝐴𝐴) = ((𝐴𝐴) ↾ V)
7 cnv0 6174 . . . 4 ∅ = ∅
85, 6, 73eqtr3i 2776 . . 3 ((𝐴𝐴) ↾ V) = ∅
93, 8sseqtri 4045 . 2 ((𝐴𝐴) ↾ 𝐵) ⊆ ∅
10 ss0b 4424 . 2 (((𝐴𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴𝐴) ↾ 𝐵) = ∅)
119, 10mpbi 230 1 ((𝐴𝐴) ↾ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  Vcvv 3488  cdif 3973  wss 3976  c0 4352  ccnv 5699  cres 5702
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 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
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 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-res 5712
This theorem is referenced by:  imanonrel  43557
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