Theorem disjALTVinidres 39237

Description: The intersection with restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)

Assertion

Ref	Expression
disjALTVinidres	⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))

Proof of Theorem disjALTVinidres

Step	Hyp	Ref	Expression
1		disjALTVid 39235	. 2 ⊢ Disj I
2		disjiminres 39232	. 2 ⊢ ( Disj I → Disj (𝑅 ∩ ( I ↾ 𝐴)))
3	1, 2	ax-mp 5	1 ⊢ Disj (𝑅 ∩ ( I ↾ 𝐴))

Syntax hints:   ∩ cin 3883   I cid 5514   ↾ cres 5622   Disj wdisjALTV 38599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-11 2170  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-coss 38881  df-cnvrefrel 38987  df-funALTV 39147  df-disjALTV 39170

This theorem is referenced by:  eqvrel1cossinidres  39274  detinidres  39279  petinidres2  39303