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Theorem resnonrel 38419
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 3767 . . . 4 𝐵 ⊆ V
2 ssres2 5584 . . . 4 (𝐵 ⊆ V → ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V))
31, 2ax-mp 5 . . 3 ((𝐴𝐴) ↾ 𝐵) ⊆ ((𝐴𝐴) ↾ V)
4 cnvnonrel 38415 . . . . 5 (𝐴𝐴) = ∅
54cnveqi 5453 . . . 4 (𝐴𝐴) =
6 cnvcnv2 5747 . . . 4 (𝐴𝐴) = ((𝐴𝐴) ↾ V)
7 cnv0 5694 . . . 4 ∅ = ∅
85, 6, 73eqtr3i 2791 . . 3 ((𝐴𝐴) ↾ V) = ∅
93, 8sseqtri 3779 . 2 ((𝐴𝐴) ↾ 𝐵) ⊆ ∅
10 ss0b 4117 . 2 (((𝐴𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴𝐴) ↾ 𝐵) = ∅)
119, 10mpbi 220 1 ((𝐴𝐴) ↾ 𝐵) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  Vcvv 3341  cdif 3713  wss 3716  c0 4059  ccnv 5266  cres 5269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-br 4806  df-opab 4866  df-xp 5273  df-rel 5274  df-cnv 5275  df-res 5279
This theorem is referenced by:  imanonrel  38420
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