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Theorem disjxrnres5 38107
Description: Disjoint range Cartesian product. (Contributed by Peter Mazsa, 25-Aug-2023.)
Assertion
Ref Expression
disjxrnres5 ( Disj (𝑅 ⋉ (𝑆𝐴)) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅𝑆) ∩ [𝑣](𝑅𝑆)) = ∅))
Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝑅,𝑣   𝑢,𝑆,𝑣

Proof of Theorem disjxrnres5
StepHypRef Expression
1 xrnres2 37763 . . 3 ((𝑅𝑆) ↾ 𝐴) = (𝑅 ⋉ (𝑆𝐴))
21disjeqi 38095 . 2 ( Disj ((𝑅𝑆) ↾ 𝐴) ↔ Disj (𝑅 ⋉ (𝑆𝐴)))
3 xrnrel 37733 . . 3 Rel (𝑅𝑆)
4 disjres 38104 . . 3 (Rel (𝑅𝑆) → ( Disj ((𝑅𝑆) ↾ 𝐴) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅𝑆) ∩ [𝑣](𝑅𝑆)) = ∅)))
53, 4ax-mp 5 . 2 ( Disj ((𝑅𝑆) ↾ 𝐴) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅𝑆) ∩ [𝑣](𝑅𝑆)) = ∅))
62, 5bitr3i 277 1 ( Disj (𝑅 ⋉ (𝑆𝐴)) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅𝑆) ∩ [𝑣](𝑅𝑆)) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 844   = wceq 1533  wral 3053  cin 3939  c0 4314  cres 5668  Rel wrel 5671  [cec 8697  cxrn 37532   Disj wdisjALTV 37567
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rmo 3368  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8701  df-xrn 37731  df-coss 37771  df-cnvrefrel 37887  df-funALTV 38042  df-disjALTV 38065
This theorem is referenced by:  disjsuc  38119
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