Theorem disjimeceqim 39812

Description: Disj implies coset-equality injectivity (domain-wise). Extracts the practical consequence of Disj: the map 𝑢 ↦ [𝑢]𝑅 is injective on dom 𝑅. This is exactly the "canonicity" property used repeatedly when turning ∃* into ∃! and when reasoning about uniqueness of representatives. (Contributed by Peter Mazsa, 3-Feb-2026.)

Assertion

Ref	Expression
disjimeceqim	⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))

Distinct variable group:   𝑢,𝑅,𝑣

Proof of Theorem disjimeceqim

Step	Hyp	Ref	Expression
1		ecdmn0 8771	. . . . . . 7 ⊢ (𝑢 ∈ dom 𝑅 ↔ [𝑢]𝑅 ≠ ∅)
2	1	biimpi 219	. . . . . 6 ⊢ (𝑢 ∈ dom 𝑅 → [𝑢]𝑅 ≠ ∅)
3		ineq2 4111	. . . . . . . 8 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑢]𝑅) = ([𝑢]𝑅 ∩ [𝑣]𝑅))
4		inidm 4123	. . . . . . . 8 ⊢ ([𝑢]𝑅 ∩ [𝑢]𝑅) = [𝑢]𝑅
5	3, 4	eqtr3di 2810	. . . . . . 7 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) = [𝑢]𝑅)
6	5	neeq1d 3014	. . . . . 6 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ ↔ [𝑢]𝑅 ≠ ∅))
7	2, 6	syl5ibrcom 250	. . . . 5 ⊢ (𝑢 ∈ dom 𝑅 → ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅))
8	7	rgen 3078	. . . 4 ⊢ ∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
9	8	rgenw 3080	. . 3 ⊢ ∀𝑣 ∈ dom 𝑅∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
10		ralcom 3290	. . 3 ⊢ (∀𝑣 ∈ dom 𝑅∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅) ↔ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅))
11	9, 10	mpbi 233	. 2 ⊢ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
12		dfdisjALTV5a 39811	. . 3 ⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))
13	12	simplbi 502	. 2 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
14		r19.26-2 3147	. . 3 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅) ∧ (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣)) ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅) ∧ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣)))
15		pm3.33 777	. . . 4 ⊢ ((([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅) ∧ (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣)) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
16	15	2ralimi 3132	. . 3 ⊢ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅) ∧ (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣)) → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
17	14, 16	sylbir 238	. 2 ⊢ ((∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅) ∧ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣)) → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
18	11, 13, 17	sylancr 599	1 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076   ∩ cin 4089  ∅c0 4278  dom cdm 5650  Rel wrel 5655  [cec 8716   Disj wdisjALTV 39227

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8720  df-coss 39509  df-cnvrefrel 39615  df-disjALTV 39798

This theorem is used by:  disjimeceqim2  39813  disjimeceqbi  39814  disjimrmoeqec  39816