Theorem disjres 38762

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 5992	. . . 4 ⊢ Rel (𝑅 ↾ 𝐴)
2		dfdisjALTV4 38734	. . . 4 ⊢ ( Disj (𝑅 ↾ 𝐴) ↔ (∀𝑥∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ∧ Rel (𝑅 ↾ 𝐴)))
3	1, 2	mpbiran2 710	. . 3 ⊢ ( Disj (𝑅 ↾ 𝐴) ↔ ∀𝑥∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥)
4		brres 5973	. . . . . . 7 ⊢ (𝑥 ∈ V → (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥)))
5	4	elv 3464	. . . . . 6 ⊢ (𝑢(𝑅 ↾ 𝐴)𝑥 ↔ (𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
6	5	mobii 2547	. . . . 5 ⊢ (∃*𝑢 𝑢(𝑅 ↾ 𝐴)𝑥 ↔ ∃*𝑢(𝑢 ∈ 𝐴 ∧ 𝑢𝑅𝑥))
7		df-rmo 3359	. . . . 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 1819	. . 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 38373	. 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 847  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∃*wmo 2537  ∀wral 3051  ∃*wrmo 3358  Vcvv 3459   ∩ cin 3925  ∅c0 4308   class class class wbr 5119   ↾ cres 5656  Rel wrel 5659  [cec 8717   Disj wdisjALTV 38233

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rmo 3359  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ec 8721  df-coss 38429  df-cnvrefrel 38545  df-disjALTV 38723

This theorem is referenced by:  disjxrnres5  38765