Theorem disjresundif 37769

Description: Lemma for ressucdifsn2 37770. (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 5993	. . . 4 ⊢ (𝑅 ↾ (𝐴 ∪ 𝐵)) = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵))
2	1	difeq1i 4110	. . 3 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵))
3		difun2 4476	. . 3 ⊢ (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
4	2, 3	eqtri 2753	. 2 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
5		disjresdif 37768	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
6	4, 5	eqtrid 2777	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1533   ∖ cdif 3937   ∪ cun 3938   ∩ cin 3939  ∅c0 4318   ↾ cres 5674

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 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5206  df-xp 5678  df-rel 5679  df-res 5684

This theorem is referenced by:  ressucdifsn2  37770