Theorem disjresundif 38745

Description: Lemma for ressucdifsn2 38986. (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 5979	. . . 4 ⊢ (𝑅 ↾ (𝐴 ∪ 𝐵)) = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵))
2	1	difeq1i 4076	. . 3 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵))
3		difun2 4435	. . 3 ⊢ (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
4	2, 3	eqtri 2785	. 2 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
5		disjresdif 38744	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
6	4, 5	eqtrid 2809	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1560   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903  ∅c0 4285   ↾ cres 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5653  df-rel 5654  df-res 5659

This theorem is referenced by:  ressucdifsn2  38986