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Theorem resnonrel 36917
 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 3588 . . . 4 𝐵 ⊆ V
2 ssres2 5345 . . . 4 (𝐵 ⊆ V → ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V))
31, 2ax-mp 5 . . 3 ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V)
4 cnvnonrel 36913 . . . . 5 (𝐴𝐴) = ∅
54cnveqi 5219 . . . 4 (𝐴𝐴) =
6 cnvcnv2 5506 . . . 4 (𝐴𝐴) = ((𝐴𝐴) ↾ V)
7 cnv0 5454 . . . 4 ∅ = ∅
85, 6, 73eqtr3i 2640 . . 3 ((𝐴𝐴) ↾ V) = ∅
93, 8sseqtri 3600 . 2 ((𝐴𝐴) ↾ 𝐵) ⊆ ∅
10 ss0b 3925 . 2 (((𝐴𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴𝐴) ↾ 𝐵) = ∅)
119, 10mpbi 219 1 ((𝐴𝐴) ↾ 𝐵) = ∅
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  ◡ccnv 5037   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-res 5050 This theorem is referenced by:  imanonrel  36918
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