Theorem rnqmapeleldisjsim

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)

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

Proof

Step	Hyp	Ref	Expression
1		rnqmap	⊢ ran QMap 𝑅 = ( dom 𝑅 / 𝑅 )
2	1	eleq1i	⊢ ( ran QMap 𝑅 ∈ ElDisjs ↔ ( dom 𝑅 / 𝑅 ) ∈ ElDisjs )
3		dmqsex	⊢ ( 𝑅 ∈ 𝑉 → ( dom 𝑅 / 𝑅 ) ∈ V )
4		eleldisjseldisj	⊢ ( ( dom 𝑅 / 𝑅 ) ∈ V → ( ( dom 𝑅 / 𝑅 ) ∈ ElDisjs ↔ ElDisj ( dom 𝑅 / 𝑅 ) ) )
5	3 4	syl	⊢ ( 𝑅 ∈ 𝑉 → ( ( dom 𝑅 / 𝑅 ) ∈ ElDisjs ↔ ElDisj ( dom 𝑅 / 𝑅 ) ) )
6	2 5	bitrid	⊢ ( 𝑅 ∈ 𝑉 → ( ran QMap 𝑅 ∈ ElDisjs ↔ ElDisj ( dom 𝑅 / 𝑅 ) ) )
7		eldisjim3	⊢ ( ElDisj ( dom 𝑅 / 𝑅 ) → ( ( [ 𝐴 ] 𝑅 ∈ ( dom 𝑅 / 𝑅 ) ∧ [ 𝐵 ] 𝑅 ∈ ( dom 𝑅 / 𝑅 ) ) → ( ( [ 𝐴 ] 𝑅 ∩ [ 𝐵 ] 𝑅 ) ≠ ∅ → [ 𝐴 ] 𝑅 = [ 𝐵 ] 𝑅 ) ) )
8		eceldmqs	⊢ ( 𝑅 ∈ 𝑉 → ( [ 𝐴 ] 𝑅 ∈ ( dom 𝑅 / 𝑅 ) ↔ 𝐴 ∈ dom 𝑅 ) )
9		eceldmqs	⊢ ( 𝑅 ∈ 𝑉 → ( [ 𝐵 ] 𝑅 ∈ ( dom 𝑅 / 𝑅 ) ↔ 𝐵 ∈ dom 𝑅 ) )
10	8 9	anbi12d	⊢ ( 𝑅 ∈ 𝑉 → ( ( [ 𝐴 ] 𝑅 ∈ ( dom 𝑅 / 𝑅 ) ∧ [ 𝐵 ] 𝑅 ∈ ( dom 𝑅 / 𝑅 ) ) ↔ ( 𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅 ) ) )
11	10	imbi1d	⊢ ( 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	⊢ ( 𝑅 ∈ 𝑉 → ( ElDisj ( dom 𝑅 / 𝑅 ) → ( ( 𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅 ) → ( ( [ 𝐴 ] 𝑅 ∩ [ 𝐵 ] 𝑅 ) ≠ ∅ → [ 𝐴 ] 𝑅 = [ 𝐵 ] 𝑅 ) ) ) )
13	6 12	sylbid	⊢ ( 𝑅 ∈ 𝑉 → ( ran QMap 𝑅 ∈ ElDisjs → ( ( 𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅 ) → ( ( [ 𝐴 ] 𝑅 ∩ [ 𝐵 ] 𝑅 ) ≠ ∅ → [ 𝐴 ] 𝑅 = [ 𝐵 ] 𝑅 ) ) ) )
14	13	3imp	⊢ ( ( 𝑅 ∈ 𝑉 ∧ ran QMap 𝑅 ∈ ElDisjs ∧ ( 𝐴 ∈ dom 𝑅 ∧ 𝐵 ∈ dom 𝑅 ) ) → ( ( [ 𝐴 ] 𝑅 ∩ [ 𝐵 ] 𝑅 ) ≠ ∅ → [ 𝐴 ] 𝑅 = [ 𝐵 ] 𝑅 ) )