Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjsuc Structured version   Visualization version   GIF version

Theorem disjsuc 37152
Description: Disjoint range Cartesian product, special case. (Contributed by Peter Mazsa, 25-Aug-2023.)
Assertion
Ref Expression
disjsuc (𝐴𝑉 → ( Disj (𝑅 ⋉ ( E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
Distinct variable groups:   𝑢,𝐴   𝑢,𝑅   𝑢,𝑉

Proof of Theorem disjsuc
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 disjsuc2 36784 . 2 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
2 df-suc 6321 . . . . . 6 suc 𝐴 = (𝐴 ∪ {𝐴})
32reseq2i 5932 . . . . 5 ( E ↾ suc 𝐴) = ( E ↾ (𝐴 ∪ {𝐴}))
43xrneq2i 36774 . . . 4 (𝑅 ⋉ ( E ↾ suc 𝐴)) = (𝑅 ⋉ ( E ↾ (𝐴 ∪ {𝐴})))
54disjeqi 37128 . . 3 ( Disj (𝑅 ⋉ ( E ↾ suc 𝐴)) ↔ Disj (𝑅 ⋉ ( E ↾ (𝐴 ∪ {𝐴}))))
6 disjxrnres5 37140 . . 3 ( Disj (𝑅 ⋉ ( E ↾ (𝐴 ∪ {𝐴}))) ↔ ∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅))
75, 6bitri 274 . 2 ( Disj (𝑅 ⋉ ( E ↾ suc 𝐴)) ↔ ∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅))
8 disjxrnres5 37140 . . 3 ( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅))
98anbi1i 624 . 2 (( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
101, 7, 93bitr4g 313 1 (𝐴𝑉 → ( Disj (𝑅 ⋉ ( E ↾ suc 𝐴)) ↔ ( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wral 3062  cun 3906  cin 3907  c0 4280  {csn 4584   E cep 5534  ccnv 5630  cres 5633  suc csuc 6317  [cec 8604  cxrn 36564   Disj wdisjALTV 36599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664  ax-reg 9486
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3351  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-eprel 5535  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7913  df-2nd 7914  df-ec 8608  df-xrn 36764  df-coss 36804  df-cnvrefrel 36920  df-funALTV 37075  df-disjALTV 37098
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator