Theorem disjres 39165

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 5970	. . . 4 ⊢ Rel (𝑅 ↾ 𝐴)
2		dfdisjALTV4 39122	. . . 4 ⊢ ( Disj (𝑅 ↾ 𝐴) ↔ (∀𝑥∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ∧ Rel (𝑅 ↾ 𝐴)))
3	1, 2	mpbiran2 711	. . 3 ⊢ ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑥∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥)
4		brres 5951	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)))
5	4	elv 3434	. . . . . 6 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
6	5	mobii 2548	. . . . 5 ⊢ (∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ↔ ∃*𝑢(𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
7		df-rmo 3342	. . . . 5 ⊢ (∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥 ↔ ∃*𝑢(𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
8	6, 7	bitr4i 278	. . . 4 ⊢ (∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ↔ ∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥)
9	8	albii 1821	. . 3 ⊢ (∀𝑥∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥)
10	3, 9	bitri 275	. 2 ⊢ ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥)
11		id 22	. . 3 ⊢ (𝑢 = 𝑣 → 𝑢 = 𝑣)
12	11	inecmo 38676	. 2 ⊢ (Rel 𝑅 → (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ ∀𝑥∃*𝑢 ∈ 𝐴 𝑢𝑅𝑥))
13	10, 12	bitr4id 290	1 ⊢ (Rel 𝑅 → ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∃*wmo 2537  ∀wral 3051  ∃*wrmo 3341  Vcvv 3429   ∩ cin 3888  ∅c0 4273   class class class wbr 5085   ↾ cres 5633  Rel wrel 5636  [cec 8641   Disj wdisjALTV 38540

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rmo 3342  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  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-ec 8645  df-coss 38822  df-cnvrefrel 38928  df-disjALTV 39111

This theorem is referenced by:  disjxrnres5  39168