Theorem rnqmapeleldisjsim 39708

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 39300	. . . . 5 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
2	1	eleq1i 2851	. . . 4 ⊢ (ran QMap 𝑅 ∈ ElDisjs ↔ (dom 𝑅 / 𝑅) ∈ ElDisjs )
3		dmqsex 39208	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 / 𝑅) ∈ V)
4		eleldisjseldisj 39675	. . . . 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 39661	. . . 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 8787	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
9		eceldmqs 8787	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ([𝐵]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐵 ∈ dom 𝑅))
10	8, 9	anbi12d 644	. . . . 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 1128	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 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450   ∩ cin 3898  ∅c0 4279  dom cdm 5648  ran crn 5649  [cec 8694   / cqs 8695   QMap cqmap 39021   ElDisjs celdisjs 39066   ElDisj weldisj 39067

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5543  df-eprel 5548  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ec 8698  df-qs 8702  df-rels 39286  df-qmap 39292  df-coss 39347  df-ssr 39424  df-cnvrefs 39451  df-cnvrefrels 39452  df-cnvrefrel 39453  df-disjss 39634  df-disjs 39635  df-disjALTV 39636  df-eldisjs 39637  df-eldisj 39638

This theorem is used by:  eldisjs6  39786