Theorem rnqmapeleldisjsim 39985

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 39577	. . . . 5 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
2	1	eleq1i 2855	. . . 4 ⊢ (ran QMap 𝑅 ∈ ElDisjs ↔ (dom 𝑅 / 𝑅) ∈ ElDisjs )
3		dmqsex 39485	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 / 𝑅) ∈ V)
4		eleldisjseldisj 39952	. . . . 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 39938	. . . 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 8833	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
9		eceldmqs 8833	. . . . . 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 2147   ≠ wne 2959  Vcvv 3454   ∩ cin 4092  ∅c0 4281  dom cdm 5655  ran crn 5656  [cec 8740   / cqs 8741   QMap cqmap 39298   ElDisjs celdisjs 39343   ElDisj weldisj 39344

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7772

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-eprel 5555  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ec 8744  df-qs 8748  df-rels 39563  df-qmap 39569  df-coss 39624  df-ssr 39701  df-cnvrefs 39728  df-cnvrefrels 39729  df-cnvrefrel 39730  df-disjss 39911  df-disjs 39912  df-disjALTV 39913  df-eldisjs 39914  df-eldisj 39915

This theorem is used by:  eldisjs6  40063