Theorem resdisj 2656

Description: A double restriction to disjoint classes is the empty set.

Assertion

Ref	Expression
resdisj	⊢ ((A ∩ B) = ∅ → ((C ↾ A) ↾ B) = ∅)

Proof of Theorem resdisj

Step	Hyp	Ref	Expression
1		xpdisj1 2653	. . . 4 ⊢ ((A ∩ B) = ∅ → ((A × V) ∩ (B × V)) = ∅)
2	1	ineq2d 1645	. . 3 ⊢ ((A ∩ B) = ∅ → (C ∩ ((A × V) ∩ (B × V))) = (C ∩ ∅))
3		in0 1722	. . 3 ⊢ (C ∩ ∅) = ∅
4	2, 3	syl6eq 1140	. 2 ⊢ ((A ∩ B) = ∅ → (C ∩ ((A × V) ∩ (B × V))) = ∅)
5		df-res 2430	. . 3 ⊢ ((C ↾ A) ↾ B) = ((C ↾ A) ∩ (B × V))
6		df-res 2430	. . . 4 ⊢ (C ↾ A) = (C ∩ (A × V))
7	6	ineq1i 1641	. . 3 ⊢ ((C ↾ A) ∩ (B × V)) = ((C ∩ (A × V)) ∩ (B × V))
8		inass 1650	. . 3 ⊢ ((C ∩ (A × V)) ∩ (B × V)) = (C ∩ ((A × V) ∩ (B × V)))
9	5, 7, 8	3eqtr 1123	. 2 ⊢ ((C ↾ A) ↾ B) = (C ∩ ((A × V) ∩ (B × V)))
10	4, 9	syl5eq 1136	1 ⊢ ((A ∩ B) = ∅ → ((C ↾ A) ↾ B) = ∅)

Syntax hints:   → wi 2   = wceq 1091  Vcvv 1348   ∩ cin 1486  ∅c0 1707   × cxp 2408   ↾ cres 2412

This theorem is referenced by:  ruclem7 4891

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-res 2430

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