Theorem disjresundif 38198

Description: Lemma for ressucdifsn2 38199. (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 6023	. . . 4 ⊢ (𝑅 ↾ (𝐴 ∪ 𝐵)) = ((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵))
2	1	difeq1i 4145	. . 3 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵))
3		difun2 4504	. . 3 ⊢ (((𝑅 ↾ 𝐴) ∪ (𝑅 ↾ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
4	2, 3	eqtri 2768	. 2 ⊢ ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵))
5		disjresdif 38197	. 2 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ 𝐴) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))
6	4, 5	eqtrid 2792	1 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝑅 ↾ (𝐴 ∪ 𝐵)) ∖ (𝑅 ↾ 𝐵)) = (𝑅 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∖ cdif 3973   ∪ cun 3974   ∩ cin 3975  ∅c0 4352   ↾ cres 5702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-opab 5229  df-xp 5706  df-rel 5707  df-res 5712

This theorem is referenced by:  ressucdifsn2  38199