Theorem disjresundif 38915

Description: Lemma for ressucdifsn2 39156. (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 5992	. . . 4 ⊢ (𝑅 ↾ (𝐴 ∪ 𝐵)) = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵))
2	1	difeq1i 4077	. . 3 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵))
3		difun2 4442	. . 3 ⊢ (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
4	2, 3	eqtri 2786	. 2 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
5		disjresdif 38914	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
6	4, 5	eqtrid 2810	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1570   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904  ∅c0 4286   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-xp 5667  df-rel 5668  df-res 5673

This theorem is referenced by:  ressucdifsn2  39156