Theorem disjimeceqim 39186

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 8690	. . . . . . 7 ⊢ (𝑢 ∈ dom 𝑅 ↔ [𝑢]𝑅 ≠ ∅)
2	1	biimpi 218	. . . . . 6 ⊢ (𝑢 ∈ dom 𝑅 → [𝑢]𝑅 ≠ ∅)
3		ineq2 4146	. . . . . . . 8 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑢]𝑅) = ([𝑢]𝑅 ∩ [𝑣]𝑅))
4		inidm 4158	. . . . . . . 8 ⊢ ([𝑢]𝑅 ∩ [𝑢]𝑅) = [𝑢]𝑅
5	3, 4	eqtr3di 2791	. . . . . . 7 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) = [𝑢]𝑅)
6	5	neeq1d 2995	. . . . . 6 ⊢ ([𝑢]𝑅 = [𝑣]𝑅 → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ ↔ [𝑢]𝑅 ≠ ∅))
7	2, 6	syl5ibrcom 249	. . . . 5 ⊢ (𝑢 ∈ dom 𝑅 → ([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅))
8	7	rgen 3057	. . . 4 ⊢ ∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
9	8	rgenw 3059	. . 3 ⊢ ∀𝑣 ∈ dom 𝑅∀𝑢 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
10		ralcom 3269	. . 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 232	. 2 ⊢ ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅)
12		dfdisjALTV5a 39185	. . 3 ⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))
13	12	simplbi 498	. 2 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
14		r19.26-2 3126	. . 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		pm3.33 771	. . . 4 ⊢ ((([𝑢]𝑅 = [𝑣]𝑅 → ([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅) ∧ (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣)) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
16	15	2ralimi 3111	. . 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 237	. 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 594	1 ⊢ ( Disj 𝑅 → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936  ∀wral 3055   ∩ cin 3884  ∅c0 4264  dom cdm 5621  Rel wrel 5626  [cec 8635   Disj wdisjALTV 38601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ec 8639  df-coss 38883  df-cnvrefrel 38989  df-disjALTV 39172

This theorem is referenced by:  disjimeceqim2  39187  disjimeceqbi  39188  disjimrmoeqec  39190