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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 e. V /\ ran QMap R e. ElDisjs /\ ( A e. dom R /\ B e. dom R ) ) -> ( ( [ A ] R i^i [ 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 e. ElDisjs <-> ( dom R /. R ) e. ElDisjs )
3 dmqsex 
 |-  ( R e. V -> ( dom R /. R ) e. _V )
4 eleldisjseldisj 
 |-  ( ( dom R /. R ) e. _V -> ( ( dom R /. R ) e. ElDisjs <-> ElDisj ( dom R /. R ) ) )
5 3 4 syl 
 |-  ( R e. V -> ( ( dom R /. R ) e. ElDisjs <-> ElDisj ( dom R /. R ) ) )
6 2 5 bitrid 
 |-  ( R e. V -> ( ran QMap R e. ElDisjs <-> ElDisj ( dom R /. R ) ) )
7 eldisjim3 
 |-  ( ElDisj ( dom R /. R ) -> ( ( [ A ] R e. ( dom R /. R ) /\ [ B ] R e. ( dom R /. R ) ) -> ( ( [ A ] R i^i [ B ] R ) =/= (/) -> [ A ] R = [ B ] R ) ) )
8 eceldmqs 
 |-  ( R e. V -> ( [ A ] R e. ( dom R /. R ) <-> A e. dom R ) )
9 eceldmqs 
 |-  ( R e. V -> ( [ B ] R e. ( dom R /. R ) <-> B e. dom R ) )
10 8 9 anbi12d 
 |-  ( R e. V -> ( ( [ A ] R e. ( dom R /. R ) /\ [ B ] R e. ( dom R /. R ) ) <-> ( A e. dom R /\ B e. dom R ) ) )
11 10 imbi1d 
 |-  ( R e. V -> ( ( ( [ A ] R e. ( dom R /. R ) /\ [ B ] R e. ( dom R /. R ) ) -> ( ( [ A ] R i^i [ B ] R ) =/= (/) -> [ A ] R = [ B ] R ) ) <-> ( ( A e. dom R /\ B e. dom R ) -> ( ( [ A ] R i^i [ B ] R ) =/= (/) -> [ A ] R = [ B ] R ) ) ) )
12 7 11 imbitrid 
 |-  ( R e. V -> ( ElDisj ( dom R /. R ) -> ( ( A e. dom R /\ B e. dom R ) -> ( ( [ A ] R i^i [ B ] R ) =/= (/) -> [ A ] R = [ B ] R ) ) ) )
13 6 12 sylbid 
 |-  ( R e. V -> ( ran QMap R e. ElDisjs -> ( ( A e. dom R /\ B e. dom R ) -> ( ( [ A ] R i^i [ B ] R ) =/= (/) -> [ A ] R = [ B ] R ) ) ) )
14 13 3imp 
 |-  ( ( R e. V /\ ran QMap R e. ElDisjs /\ ( A e. dom R /\ B e. dom R ) ) -> ( ( [ A ] R i^i [ B ] R ) =/= (/) -> [ A ] R = [ B ] R ) )