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Metamath Proof Explorer

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 ) )