Theorem disjiminres 39752

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 39750	. 2 ⊢ ( Disj 𝑆 → Disj (𝑆 ↾ 𝐴))
2		disjimin 39751	. 2 ⊢ ( Disj (𝑆 ↾ 𝐴) → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))
3	1, 2	syl 18	1 ⊢ ( Disj 𝑆 → Disj (𝑅 ∩ (𝑆 ↾ 𝐴)))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3898   ↾ cres 5653   Disj wdisjALTV 39119

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-11 2194  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-ral 3078  df-rex 3088  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-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-coss 39401  df-cnvrefrel 39507  df-funALTV 39667  df-disjALTV 39690

This theorem is used by:  disjALTVinidres  39757