Theorem eldisjdmqsim2 39198

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 39197	. . 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 8728	. . . . 5 ⊢ (𝑅 ∈ Rels → ([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑢 ∈ dom 𝑅))
3		eceldmqs 8728	. . . . 5 ⊢ (𝑅 ∈ Rels → ([𝑣]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝑣 ∈ dom 𝑅))
4	2, 3	anbi12d 639	. . . 4 ⊢ (𝑅 ∈ Rels → (([𝑢]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝑣]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ (𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅)))
5	4	imbi1d 343	. . 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 246	. 2 ⊢ (𝑅 ∈ Rels → ( ElDisj (dom 𝑅 / 𝑅) → ((𝑢 ∈ dom 𝑅 ∧ 𝑣 ∈ dom 𝑅) → (([𝑢]𝑅 ∩ [𝑣]𝑅) ≠ ∅ → [𝑢]𝑅 = [𝑣]𝑅))))
7	6	impcom 409	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 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936   ∩ cin 3884  ∅c0 4264  dom cdm 5621  [cec 8635   / cqs 8636   Rels crels 38567   ElDisj weldisj 38603

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  ax-un 7682

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-uni 4842  df-br 5076  df-opab 5138  df-id 5516  df-eprel 5521  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-qs 8643  df-coss 38883  df-cnvrefrel 38989  df-disjALTV 39172  df-eldisj 39174

This theorem is referenced by: (None)