Theorem eldisjs6 39873

Description: Elementhood in the class of disjoints. A relation 𝑅 is in Disjs iff:

it is relation-typed, and

its quotient-map QMap 𝑅 is itself disjoint, and

its quotient-carrier ran QMap 𝑅 = (dom 𝑅 / 𝑅) lies in ElDisjs (element-disjoint carriers).

This is the central "stability-by-decomposition" theorem for Disjs: it explains why Disjs is internally well-behaved without adding an external stability clause. It is the exact template that PetParts imitates: for pet 39898, the analogue of "map layer" is the disjointness of the lifted span, the analogue of "carrier layer" is the block-lift fixpoint (BlockLiftFix), and then adds external grade stability (SucMap ShiftStable) which Disjs does not need. (Contributed by Peter Mazsa, 16-Feb-2026.)

Assertion

Ref	Expression
eldisjs6	⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))

Proof of Theorem eldisjs6

Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldisjsim2 39868	. . 3 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )
2		eldisjsim4 39871	. . 3 ⊢ (𝑅 ∈ Disjs → ran QMap 𝑅 ∈ ElDisjs )
3		eldisjsim5 39872	. . 3 ⊢ (𝑅 ∈ Disjs → QMap 𝑅 ∈ Disjs )
4	1, 2, 3	jca32 525	. 2 ⊢ (𝑅 ∈ Disjs → (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))
5		rnqmapeleldisjsim 39795	. . . . . . 7 ⊢ ((𝑅 ∈ Rels ∧ ran QMap 𝑅 ∈ ElDisjs ∧ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅))
6	5	3adant2r 1198	. . . . . 6 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs ) ∧ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅))
7		qmapeldisjsim 39793	. . . . . . 7 ⊢ ((𝑅 ∈ Rels ∧ QMap 𝑅 ∈ Disjs ∧ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
8	7	3adant2l 1197	. . . . . 6 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs ) ∧ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
9	6, 8	syld 48	. . . . 5 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs ) ∧ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
10	9	3expia 1139	. . . 4 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣)))
11	10	ralrimivv 3203	. . 3 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )) → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
12		elrelsrelim 39376	. . . . . 6 ⊢ (𝑅 ∈ Rels → Rel 𝑅)
13		dfdisjALTV5a 39736	. . . . . . 7 ⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))
14	13	simplbi2com 508	. . . . . 6 ⊢ (Rel 𝑅 → (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) → Disj 𝑅))
15	12, 14	syl 18	. . . . 5 ⊢ (𝑅 ∈ Rels → (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) → Disj 𝑅))
16		eldisjsdisj 39757	. . . . 5 ⊢ (𝑅 ∈ Rels → (𝑅 ∈ Disjs ↔ Disj 𝑅))
17	15, 16	sylibrd 262	. . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) → 𝑅 ∈ Disjs ))
18	17	adantr 486	. . 3 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )) → (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) → 𝑅 ∈ Disjs ))
19	11, 18	mpd 16	. 2 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )) → 𝑅 ∈ Disjs )
20	4, 19	impbii 212	1 ⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076   ∩ cin 4089  ∅c0 4278  dom cdm 5650  ran crn 5651  Rel wrel 5655  [cec 8715   QMap cqmap 39108   Rels crels 39118   Disjs cdisjs 39151   Disj wdisjALTV 39152   ElDisjs celdisjs 39153

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-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

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-eu 2594  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-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-eprel 5550  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-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ec 8719  df-qs 8723  df-rels 39373  df-qmap 39379  df-coss 39434  df-ssr 39511  df-refrel 39525  df-cnvrefs 39538  df-cnvrefrels 39539  df-cnvrefrel 39540  df-symrel 39557  df-trrel 39591  df-eqvrel 39602  df-funALTV 39700  df-disjss 39721  df-disjs 39722  df-disjALTV 39723  df-eldisjs 39724  df-eldisj 39725

This theorem is used by:  eldisjs7  39874  dfdisjs6  39875