Theorem disjiminres 38860

Description: Disjointness condition for intersection with restriction. (Contributed by Peter Mazsa, 27-Sep-2021.)

Assertion

Ref	Expression
disjiminres	⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))

Proof of Theorem disjiminres

Step	Hyp	Ref	Expression
1		disjimres 38858	. 2 ⊢ ( Disj 𝑆 → Disj (𝑆 ↾ 𝐴))
2		disjimin 38859	. 2 ⊢ ( Disj (𝑆 ↾ 𝐴) → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))
3	1, 2	syl 17	1 ⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))

Syntax hints:   → wi 4   ∩ cin 3896   ↾ cres 5616   Disj wdisjALTV 38266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-11 2160  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-coss 38523  df-cnvrefrel 38629  df-funALTV 38790  df-disjALTV 38813

This theorem is referenced by:  disjALTVinidres  38865