Theorem disjresdif 37621

Description: The difference between restrictions to disjoint is the first restriction. (Contributed by Peter Mazsa, 24-Jul-2024.)

Assertion

Ref	Expression
disjresdif	⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))

Proof of Theorem disjresdif

Step	Hyp	Ref	Expression
1		disjresdisj 37620	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵)) = ∅)
2		disjdif2 4474	. 2 ⊢ (((𝑅 ↾ 𝐴) ∩ (𝑅 ↾ 𝐵)) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
3	1, 2	syl 17	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1533   ∖ cdif 3940   ∩ cin 3942  ∅c0 4317   ↾ cres 5671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-opab 5204  df-xp 5675  df-rel 5676  df-res 5681

This theorem is referenced by:  disjresundif  37622