Theorem disjresundif 38224

Description: Lemma for ressucdifsn2 38225. (Contributed by Peter Mazsa, 24-Jul-2024.)

Assertion

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

Proof of Theorem disjresundif

Step	Hyp	Ref	Expression
1		resundi 6014	. . . 4 ⊢ (𝑅 ↾ (𝐴 ∪ 𝐵)) = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵))
2	1	difeq1i 4132	. . 3 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵))
3		difun2 4487	. . 3 ⊢ (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
4	2, 3	eqtri 2763	. 2 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
5		disjresdif 38223	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
6	4, 5	eqtrid 2787	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962  ∅c0 4339   ↾ cres 5691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696  df-res 5701

This theorem is referenced by:  ressucdifsn2  38225