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Theorem disjres 38745
Description: Disjoint restriction. (Contributed by Peter Mazsa, 25-Aug-2023.)
Assertion
Ref Expression
disjres (Rel 𝑅 → ( Disj (𝑅𝐴) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅)))
Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝑅,𝑣

Proof of Theorem disjres
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 relres 6023 . . . 4 Rel (𝑅𝐴)
2 dfdisjALTV4 38717 . . . 4 ( Disj (𝑅𝐴) ↔ (∀𝑥∃*𝑢 𝑢(𝑅𝐴)𝑥 ∧ Rel (𝑅𝐴)))
31, 2mpbiran2 710 . . 3 ( Disj (𝑅𝐴) ↔ ∀𝑥∃*𝑢 𝑢(𝑅𝐴)𝑥)
4 brres 6004 . . . . . . 7 (𝑥 ∈ V → (𝑢(𝑅𝐴)𝑥 ↔ (𝑢𝐴𝑢𝑅𝑥)))
54elv 3485 . . . . . 6 (𝑢(𝑅𝐴)𝑥 ↔ (𝑢𝐴𝑢𝑅𝑥))
65mobii 2548 . . . . 5 (∃*𝑢 𝑢(𝑅𝐴)𝑥 ↔ ∃*𝑢(𝑢𝐴𝑢𝑅𝑥))
7 df-rmo 3380 . . . . 5 (∃*𝑢𝐴 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢𝐴𝑢𝑅𝑥))
86, 7bitr4i 278 . . . 4 (∃*𝑢 𝑢(𝑅𝐴)𝑥 ↔ ∃*𝑢𝐴 𝑢𝑅𝑥)
98albii 1819 . . 3 (∀𝑥∃*𝑢 𝑢(𝑅𝐴)𝑥 ↔ ∀𝑥∃*𝑢𝐴 𝑢𝑅𝑥)
103, 9bitri 275 . 2 ( Disj (𝑅𝐴) ↔ ∀𝑥∃*𝑢𝐴 𝑢𝑅𝑥)
11 id 22 . . 3 (𝑢 = 𝑣𝑢 = 𝑣)
1211inecmo 38356 . 2 (Rel 𝑅 → (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ ∀𝑥∃*𝑢𝐴 𝑢𝑅𝑥))
1310, 12bitr4id 290 1 (Rel 𝑅 → ( Disj (𝑅𝐴) ↔ ∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 848  wal 1538   = wceq 1540  wcel 2108  ∃*wmo 2538  wral 3061  ∃*wrmo 3379  Vcvv 3480  cin 3950  c0 4333   class class class wbr 5143  cres 5687  Rel wrel 5690  [cec 8743   Disj wdisjALTV 38216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-coss 38412  df-cnvrefrel 38528  df-disjALTV 38706
This theorem is referenced by:  disjxrnres5  38748
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