Theorem resnonrel 44577

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 3955	. . . 4 ⊢ 𝐵 ⊆ V
2		ssres2 5995	. . . 4 ⊢ (𝐵 ⊆ V → ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V))
3	1, 2	ax-mp 5	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ((𝐴 ∖ ◡◡𝐴) ↾ V)
4		cnvnonrel 44573	. . . . 5 ⊢ ◡(𝐴 ∖ ◡◡𝐴) = ∅
5	4	cnveqi 5852	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ◡∅
6		cnvcnv2 6185	. . . 4 ⊢ ◡◡(𝐴 ∖ ◡◡𝐴) = ((𝐴 ∖ ◡◡𝐴) ↾ V)
7		cnv0 5861	. . . 4 ⊢ ◡∅ = ∅
8	5, 6, 7	3eqtr3i 2792	. . 3 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ V) = ∅
9	3, 8	sseqtri 3979	. 2 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅
10		ss0b 4351	. 2 ⊢ (((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) ⊆ ∅ ↔ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅)
11	9, 10	mpbi 233	1 ⊢ ((𝐴 ∖ ◡◡𝐴) ↾ 𝐵) = ∅

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ^◡ccnv 5650   ↾ cres 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-res 5663

This theorem is used by:  imanonrel  44578