Theorem disjiminres 39132

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 39130	. 2 ⊢ ( Disj 𝑆 → Disj (𝑆 ↾ 𝐴))
2		disjimin 39131	. 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 3902   ↾ cres 5636   Disj wdisjALTV 38499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-coss 38781  df-cnvrefrel 38887  df-funALTV 39047  df-disjALTV 39070

This theorem is referenced by:  disjALTVinidres  39137