Theorem disjres 39493

Description: Disjoint restriction. (Contributed by Peter Mazsa, 25-Aug-2023.)

Assertion

Ref	Expression
disjres	⊢ (Rel 𝑅 → ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (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.

Step	Hyp	Ref	Expression
1		relres 6004	. . . 4 ⊢ Rel (𝑅 ↾ 𝐴)
2		dfdisjALTV4 39450	. . . 4 ⊢ ( Disj (𝑅 ↾ 𝐴) ↔ (∀𝑥∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ∧ Rel (𝑅 ↾ 𝐴)))
3	1, 2	mpbiran2 722	. . 3 ⊢ ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑥∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥)
4		brres 5985	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)))
5	4	elv 3460	. . . . . 6 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
6	5	mobii 2576	. . . . 5 ⊢ (∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ↔ ∃*𝑢(𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
7		df-rmo 3369	. . . . 5 ⊢ (∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
8	6, 7	bitr4i 281	. . . 4 ⊢ (∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ↔ ∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥)
9	8	albii 1849	. . 3 ⊢ (∀𝑥∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥)
10	3, 9	bitri 278	. 2 ⊢ ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥)
11		id 23	. . 3 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
12	11	inecmo 39004	. 2 ⊢ (Rel 𝑅 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥))
13	10, 12	bitr4id 293	1 ⊢ (Rel 𝑅 → ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860  ∀wal 1568   = wceq 1570   ∈ wcel 2143  ∃*wmo 2565  ∀wral 3079  ∃*wrmo 3368  Vcvv 3455   ∩ cin 3904  ∅c0 4286   class class class wbr 5109   ↾ cres 5663  Rel wrel 5666  [cec 8688   Disj wdisjALTV 38868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-coss 39150  df-cnvrefrel 39256  df-disjALTV 39439

This theorem is referenced by:  disjxrnres5  39496