Theorem disjimeceqim 39539

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 8752	. . . . . . 7 ⊢ (𝑢 ∈ dom 𝑅 ↔ [𝑢]𝑅 ≠ ∅)
2	1	biimpi 219	. . . . . 6 ⊢ (𝑢 ∈ dom 𝑅 → [𝑢]𝑅 ≠ ∅)
3		ineq2 4163	. . . . . . . 8 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑢]𝑅) = ([𝑢]𝑅 ∩ [𝑣]𝑅))
4		inidm 4175	. . . . . . . 8 ⊢ ([𝑢]𝑅 ∩ [𝑢]𝑅) = [𝑢]𝑅
5	3, 4	eqtr3di 2812	. . . . . . 7 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) = [𝑢]𝑅)
6	5	neeq1d 3016	. . . . . 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 3080	. . . 4 ⊢ ∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
9	8	rgenw 3082	. . 3 ⊢ ∀𝑣 ∈ dom 𝑅∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
10		ralcom 3292	. . 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 39538	. . 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 3149	. . 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 3134	. . 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 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078   ∩ cin 3901  ∅c0 4282  dom cdm 5659  Rel wrel 5664  [cec 8697   Disj wdisjALTV 38954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8701  df-coss 39236  df-cnvrefrel 39342  df-disjALTV 39525

This theorem is used by:  disjimeceqim2  39540  disjimeceqbi  39541  disjimrmoeqec  39543