Theorem rnqmapeleldisjsim 39183

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 38775	. . . . 5 ⊢ ran QMap 𝑅 = (dom 𝑅 / 𝑅)
2	1	eleq1i 2827	. . . 4 ⊢ (ran QMap 𝑅 ∈ ElDisjs ↔ (dom 𝑅 / 𝑅) ∈ ElDisjs )
3		dmqsex 38683	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (dom 𝑅 / 𝑅) ∈ V)
4		eleldisjseldisj 39150	. . . . 5 ⊢ ((dom 𝑅 / 𝑅) ∈ V → ((dom 𝑅 / 𝑅) ∈ ElDisjs ↔ ElDisj (dom 𝑅 / 𝑅)))
5	3, 4	syl 17	. . . 4 ⊢ (𝑅 ∈ 𝑉 → ((dom 𝑅 / 𝑅) ∈ ElDisjs ↔ ElDisj (dom 𝑅 / 𝑅)))
6	2, 5	bitrid 283	. . 3 ⊢ (𝑅 ∈ 𝑉 → (ran QMap 𝑅 ∈ ElDisjs ↔ ElDisj (dom 𝑅 / 𝑅)))
7		eldisjim3 39136	. . . 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 8734	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐴 ∈ dom 𝑅))
9		eceldmqs 8734	. . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ([𝐵]𝑅 ∈ (dom 𝑅 / 𝑅) ↔ 𝐵 ∈ dom 𝑅))
10	8, 9	anbi12d 633	. . . . 5 ⊢ (𝑅 ∈ 𝑉 → (([𝐴]𝑅 ∈ (dom 𝑅 / 𝑅) ∧ [𝐵]𝑅 ∈ (dom 𝑅 / 𝑅)) ↔ (𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅)))
11	10	imbi1d 341	. . . 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 244	. . 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 240	. 2 ⊢ (𝑅 ∈ 𝑉 → (ran QMap 𝑅 ∈ ElDisjs → ((𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅) → (([𝐴]𝑅 ∩ [𝐵]𝑅) ≠ ∅ → [𝐴]𝑅 = [𝐵]𝑅))))
14	13	3imp 1111	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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  Vcvv 3429   ∩ cin 3888  ∅c0 4273  dom cdm 5631  ran crn 5632  [cec 8641   / cqs 8642   QMap cqmap 38496   ElDisjs celdisjs 38541   ElDisj weldisj 38542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-eprel 5531  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8645  df-qs 8649  df-rels 38761  df-qmap 38767  df-coss 38822  df-ssr 38899  df-cnvrefs 38926  df-cnvrefrels 38927  df-cnvrefrel 38928  df-disjss 39109  df-disjs 39110  df-disjALTV 39111  df-eldisjs 39112  df-eldisj 39113

This theorem is referenced by:  eldisjs6  39261