Theorem rnqmapeleldisjsim 39539

Description: Element-disjointness of the quotient carrier forces coset disjointness. Supplies the "cosets don't overlap unless equal" direction, but expressed via ran QMap 𝑅 (the quotient carrier) and ElDisjs. This is the structural reason Disjs needs a "carrier disjointness" level distinct from the "unique representatives" level. (Contributed by Peter Mazsa, 16-Feb-2026.)

Assertion

Ref	Expression
rnqmapeleldisjsim	⊢ ((𝑅 ∈ 𝑉 ∧ ran QMap 𝑅 ∈ ElDisjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅))

Proof of Theorem rnqmapeleldisjsim

Step	Hyp	Ref	Expression
1		rnqmap 39131	. . . . 5 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
2	1	eleq1i 2853	. . . 4 ⊢ (ran QMap 𝑅 ∈ ElDisjs ↔ (dom 𝑅 / 𝑅) ∈ ElDisjs )
3		dmqsex 39039	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 / 𝑅) ∈ V)
4		eleldisjseldisj 39506	. . . . 5 ⊢ ((dom 𝑅 / 𝑅) ∈ V → ((dom 𝑅 / 𝑅) ∈ ElDisjs ↔ ElDisj (dom 𝑅 / 𝑅)))
5	3, 4	syl 18	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((dom 𝑅 / 𝑅) ∈ ElDisjs ↔ ElDisj (dom 𝑅 / 𝑅)))
6	2, 5	bitrid 286	. . 3 ⊢ (𝑅 ∈ 𝑉 → (ran QMap 𝑅 ∈ ElDisjs ↔ ElDisj (dom 𝑅 / 𝑅)))
7		eldisjim3 39492	. . . 4 ⊢ ( ElDisj (dom 𝑅 / 𝑅) → (([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝐵]𝑅 ∈ (dom 𝑅 / 𝑅)) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅)))
8		eceldmqs 8783	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
9		eceldmqs 8783	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ([𝐵]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐵 ∈ dom 𝑅))
10	8, 9	anbi12d 643	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝐵]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
11	10	imbi1d 344	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝐵]𝑅 ∈ (dom 𝑅 / 𝑅)) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅)) ↔ ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅))))
12	7, 11	imbitrid 247	. . 3 ⊢ (𝑅 ∈ 𝑉 → ( ElDisj (dom 𝑅 / 𝑅) → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅))))
13	6, 12	sylbid 243	. 2 ⊢ (𝑅 ∈ 𝑉 → (ran QMap 𝑅 ∈ ElDisjs → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅))))
14	13	3imp 1127	1 ⊢ ((𝑅 ∈ 𝑉 ∧ ran QMap 𝑅 ∈ ElDisjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454   ∩ cin 3903  ∅c0 4285  dom cdm 5660  ran crn 5661  [cec 8690   / cqs 8691   QMap cqmap 38852   ElDisjs celdisjs 38897   ElDisj weldisj 38898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-eprel 5560  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-ec 8694  df-qs 8698  df-rels 39117  df-qmap 39123  df-coss 39178  df-ssr 39255  df-cnvrefs 39282  df-cnvrefrels 39283  df-cnvrefrel 39284  df-disjss 39465  df-disjs 39466  df-disjALTV 39467  df-eldisjs 39468  df-eldisj 39469

This theorem is used by:  eldisjs6  39617