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Theorem reldisjun 6052
Description: Split a relation into two disjoint parts based on its domain. (Contributed by Thierry Arnoux, 9-Oct-2023.)
Assertion
Ref Expression
reldisjun ((Rel 𝑅 ∧ dom 𝑅 = (𝐴𝐵) ∧ (𝐴𝐵) = ∅) → 𝑅 = ((𝑅𝐴) ∪ (𝑅𝐵)))

Proof of Theorem reldisjun
StepHypRef Expression
1 reseq2 5995 . . 3 (dom 𝑅 = (𝐴𝐵) → (𝑅 ↾ dom 𝑅) = (𝑅 ↾ (𝐴𝐵)))
213ad2ant2 1133 . 2 ((Rel 𝑅 ∧ dom 𝑅 = (𝐴𝐵) ∧ (𝐴𝐵) = ∅) → (𝑅 ↾ dom 𝑅) = (𝑅 ↾ (𝐴𝐵)))
3 resdm 6046 . . 3 (Rel 𝑅 → (𝑅 ↾ dom 𝑅) = 𝑅)
433ad2ant1 1132 . 2 ((Rel 𝑅 ∧ dom 𝑅 = (𝐴𝐵) ∧ (𝐴𝐵) = ∅) → (𝑅 ↾ dom 𝑅) = 𝑅)
5 resundi 6014 . . 3 (𝑅 ↾ (𝐴𝐵)) = ((𝑅𝐴) ∪ (𝑅𝐵))
65a1i 11 . 2 ((Rel 𝑅 ∧ dom 𝑅 = (𝐴𝐵) ∧ (𝐴𝐵) = ∅) → (𝑅 ↾ (𝐴𝐵)) = ((𝑅𝐴) ∪ (𝑅𝐵)))
72, 4, 63eqtr3d 2783 1 ((Rel 𝑅 ∧ dom 𝑅 = (𝐴𝐵) ∧ (𝐴𝐵) = ∅) → 𝑅 = ((𝑅𝐴) ∪ (𝑅𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  cun 3961  cin 3962  c0 4339  dom cdm 5689  cres 5691  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701
This theorem is referenced by:  fressupp  32703  cycpmconjslem2  33158  evlselvlem  42573
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