Theorem resnonrel 44318

Description: A restriction of the non-relation part of a class is empty. (Contributed by RP, 22-Oct-2020.)

Assertion

Ref	Expression
resnonrel	⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅

Proof of Theorem resnonrel

Step	Hyp	Ref	Expression
1		ssv 3961	. . . 4 ⊢ 𝐵 ⊆ V
2		ssres2 6003	. . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)
4		cnvnonrel 44314	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
5	4	cnveqi 5860	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅
6		cnvcnv2 6191	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V)
7		cnv0 5869	. . . 4 ⊢ ◡∅ = ∅
8	5, 6, 7	3eqtr3i 2794	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅
9	3, 8	sseqtri 3985	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅
10		ss0b 4358	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅)
11	9, 10	mpbi 233	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅

Syntax hints:   = wceq 1570  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ∅c0 4286  ^◡ccnv 5660   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-cnv 5669  df-res 5673

This theorem is referenced by:  imanonrel  44319