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Theorem disjresundif 38245
Description: Lemma for ressucdifsn2 38246. (Contributed by Peter Mazsa, 24-Jul-2024.)
Assertion
Ref Expression
disjresundif ((𝐴𝐵) = ∅ → ((𝑅 ↾ (𝐴𝐵)) ∖ (𝑅𝐵)) = (𝑅𝐴))

Proof of Theorem disjresundif
StepHypRef Expression
1 resundi 6010 . . . 4 (𝑅 ↾ (𝐴𝐵)) = ((𝑅𝐴) ∪ (𝑅𝐵))
21difeq1i 4121 . . 3 ((𝑅 ↾ (𝐴𝐵)) ∖ (𝑅𝐵)) = (((𝑅𝐴) ∪ (𝑅𝐵)) ∖ (𝑅𝐵))
3 difun2 4480 . . 3 (((𝑅𝐴) ∪ (𝑅𝐵)) ∖ (𝑅𝐵)) = ((𝑅𝐴) ∖ (𝑅𝐵))
42, 3eqtri 2764 . 2 ((𝑅 ↾ (𝐴𝐵)) ∖ (𝑅𝐵)) = ((𝑅𝐴) ∖ (𝑅𝐵))
5 disjresdif 38244 . 2 ((𝐴𝐵) = ∅ → ((𝑅𝐴) ∖ (𝑅𝐵)) = (𝑅𝐴))
64, 5eqtrid 2788 1 ((𝐴𝐵) = ∅ → ((𝑅 ↾ (𝐴𝐵)) ∖ (𝑅𝐵)) = (𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  cdif 3947  cun 3948  cin 3949  c0 4332  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-xp 5690  df-rel 5691  df-res 5696
This theorem is referenced by:  ressucdifsn2  38246
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