Theorem rnqmapeleldisjsim 39461

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 39053	. . . . 5 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
2	1	eleq1i 2861	. . . 4 ⊢ (ran QMap 𝑅 ∈ ElDisjs ↔ (dom 𝑅 / 𝑅) ∈ ElDisjs )
3		dmqsex 38961	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 / 𝑅) ∈ V)
4		eleldisjseldisj 39428	. . . . 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 39414	. . . 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 8788	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
9		eceldmqs 8788	. . . . . 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 1126	1 ⊢ ((𝑅 ∈ 𝑉 ∧ ran QMap 𝑅 ∈ ElDisjs ∧ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150   ≠ wne 2965  Vcvv 3462   ∩ cin 3912  ∅c0 4294  dom cdm 5665  ran crn 5666  [cec 8695   / cqs 8696   QMap cqmap 38774   ElDisjs celdisjs 38819   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-rep 5243  ax-sep 5262  ax-pow 5340  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-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  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-rels 39039  df-qmap 39045  df-coss 39100  df-ssr 39177  df-cnvrefs 39204  df-cnvrefrels 39205  df-cnvrefrel 39206  df-disjss 39387  df-disjs 39388  df-disjALTV 39389  df-eldisjs 39390  df-eldisj 39391

This theorem is referenced by:  eldisjs6  39539