Theorem disjimeceqim 38974

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 8688	. . . . . . 7 ⊢ (𝑢 ∈ dom 𝑅 ↔ [𝑢]𝑅 ≠ ∅)
2	1	biimpi 216	. . . . . 6 ⊢ (𝑢 ∈ dom 𝑅 → [𝑢]𝑅 ≠ ∅)
3		ineq2 4165	. . . . . . . 8 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑢]𝑅) = ([𝑢]𝑅 ∩ [𝑣]𝑅))
4		inidm 4178	. . . . . . . 8 ⊢ ([𝑢]𝑅 ∩ [𝑢]𝑅) = [𝑢]𝑅
5	3, 4	eqtr3di 2785	. . . . . . 7 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) = [𝑢]𝑅)
6	5	neeq1d 2990	. . . . . 6 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ ↔ [𝑢]𝑅 ≠ ∅))
7	2, 6	syl5ibrcom 247	. . . . 5 ⊢ (𝑢 ∈ dom 𝑅 → ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅))
8	7	rgen 3052	. . . 4 ⊢ ∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
9	8	rgenw 3054	. . 3 ⊢ ∀𝑣 ∈ dom 𝑅∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
10		ralcom 3263	. . 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 230	. 2 ⊢ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
12		dfdisjALTV5a 38973	. . 3 ⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))
13	12	simplbi 497	. 2 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
14		r19.26-2 3120	. . 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		pm3.33 765	. . . 4 ⊢ ((([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅) ∧ (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣)) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
16	15	2ralimi 3105	. . 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 235	. 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 588	1 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2931  ∀wral 3050   ∩ cin 3899  ∅c0 4284  dom cdm 5623  Rel wrel 5628  [cec 8633   Disj wdisjALTV 38389

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 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ec 8637  df-coss 38671  df-cnvrefrel 38777  df-disjALTV 38960

This theorem is referenced by:  disjimeceqim2  38975  disjimeceqbi  38976  disjimrmoeqec  38978