Theorem eldisjs6 39143

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 39168, 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 39138	. . 3 ⊢ (𝑅 ∈ Disjs → 𝑅 ∈ Rels )
2		eldisjsim4 39141	. . 3 ⊢ (𝑅 ∈ Disjs → ran QMap 𝑅 ∈ ElDisjs )
3		eldisjsim5 39142	. . 3 ⊢ (𝑅 ∈ Disjs → QMap 𝑅 ∈ Disjs )
4	1, 2, 3	jca32 515	. 2 ⊢ (𝑅 ∈ Disjs → (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))
5		rnqmapeleldisjsim 39065	. . . . . . 7 ⊢ ((𝑅 ∈ Rels ∧ ran QMap 𝑅 ∈ ElDisjs ∧ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅))
6	5	3adant2r 1181	. . . . . 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 39063	. . . . . . 7 ⊢ ((𝑅 ∈ Rels ∧ QMap 𝑅 ∈ Disjs ∧ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)) → ([𝑢]𝑅 = [𝑣]𝑅 → 𝑢 = 𝑣))
8	7	3adant2l 1180	. . . . . 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 47	. . . . 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 1122	. . . 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 3178	. . 3 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )) → ∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣))
12		elrelsrelim 38646	. . . . . 6 ⊢ (𝑅 ∈ Rels → Rel 𝑅)
13		dfdisjALTV5a 39006	. . . . . . 7 ⊢ ( Disj 𝑅 ↔ (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) ∧ Rel 𝑅))
14	13	simplbi2com 502	. . . . . 6 ⊢ (Rel 𝑅 → (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) → Disj 𝑅))
15	12, 14	syl 17	. . . . 5 ⊢ (𝑅 ∈ Rels → (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) → Disj 𝑅))
16		eldisjsdisj 39027	. . . . 5 ⊢ (𝑅 ∈ Rels → (𝑅 ∈ Disjs ↔ Disj 𝑅))
17	15, 16	sylibrd 259	. . . 4 ⊢ (𝑅 ∈ Rels → (∀𝑢 ∈ dom 𝑅∀𝑣 ∈ dom 𝑅(([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → 𝑢 = 𝑣) → 𝑅 ∈ Disjs ))
18	17	adantr 480	. . 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 15	. 2 ⊢ ((𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )) → 𝑅 ∈ Disjs )
20	4, 19	impbii 209	1 ⊢ (𝑅 ∈ Disjs ↔ (𝑅 ∈ Rels ∧ (ran QMap 𝑅 ∈ ElDisjs ∧ QMap 𝑅 ∈ Disjs )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052   ∩ cin 3901  ∅c0 4286  dom cdm 5625  ran crn 5626  Rel wrel 5630  [cec 8635   QMap cqmap 38378   Rels crels 38388   Disjs cdisjs 38421   Disj wdisjALTV 38422   ElDisjs celdisjs 38423

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 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ec 8639  df-qs 8643  df-rels 38643  df-qmap 38649  df-coss 38704  df-ssr 38781  df-refrel 38795  df-cnvrefs 38808  df-cnvrefrels 38809  df-cnvrefrel 38810  df-symrel 38827  df-trrel 38861  df-eqvrel 38872  df-funALTV 38970  df-disjss 38991  df-disjs 38992  df-disjALTV 38993  df-eldisjs 38994  df-eldisj 38995

This theorem is referenced by:  eldisjs7  39144  dfdisjs6  39145