Theorem eldisjdmqsim2 39415

Description: ElDisj of quotient implies coset-disjointness (domain form). Converts element-disjointness of the quotient carrier into a usable "cosets don't overlap unless equal" rule. (Contributed by Peter Mazsa, 10-Feb-2026.)

Assertion

Ref	Expression
eldisjdmqsim2	⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))

Proof of Theorem eldisjdmqsim2

Step	Hyp	Ref	Expression
1		eldisjim3 39414	. . 3 ⊢ ( ElDisj (dom 𝑅 / 𝑅) → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))
2		eceldmqs 8788	. . . . 5 ⊢ (𝑅 ∈ Rels → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑢 ∈ dom 𝑅))
3		eceldmqs 8788	. . . . 5 ⊢ (𝑅 ∈ Rels → ([𝑣]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑣 ∈ dom 𝑅))
4	2, 3	anbi12d 643	. . . 4 ⊢ (𝑅 ∈ Rels → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)))
5	4	imbi1d 344	. . 3 ⊢ (𝑅 ∈ Rels → ((([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)) ↔ ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅))))
6	1, 5	imbitrid 247	. 2 ⊢ (𝑅 ∈ Rels → ( ElDisj (dom 𝑅 / 𝑅) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅))))
7	6	impcom 412	1 ⊢ (( ElDisj (dom 𝑅 / 𝑅) ∧ 𝑅 ∈ Rels ) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2150   ≠ wne 2965   ∩ cin 3912  ∅c0 4294  dom cdm 5665  [cec 8695   / cqs 8696   Rels crels 38784   ElDisj weldisj 38820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rmo 3376  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5560  df-eprel 5565  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ec 8699  df-qs 8703  df-coss 39100  df-cnvrefrel 39206  df-disjALTV 39389  df-eldisj 39391

This theorem is referenced by: (None)