Metamath Proof Explorer

Theorem qmapeldisjsim

Description: Injectivity of coset map from QMap being disjoint (implication form): under the Disjs condition on QMap R , the coset assignment is injective on dom R . (Contributed by Peter Mazsa, 16-Feb-2026)

Ref Expression
Assertion qmapeldisjsim 
|- ( ( R e. V /\ QMap R e. Disjs /\ ( A e. dom R /\ B e. dom R ) ) -> ( [ A ] R = [ B ] R -> A = B ) )

Proof

Step Hyp Ref Expression
1 qmapeldisjs 
 |-  ( R e. V -> ( QMap R e. Disjs <-> Disj QMap R ) )
2 disjimeceqim2 
 |-  ( Disj QMap R -> ( ( A e. dom QMap R /\ B e. dom QMap R ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) )
3 dmqmap 
 |-  ( R e. V -> dom QMap R = dom R )
4 3 eleq2d 
 |-  ( R e. V -> ( A e. dom QMap R <-> A e. dom R ) )
5 3 eleq2d 
 |-  ( R e. V -> ( B e. dom QMap R <-> B e. dom R ) )
6 4 5 anbi12d 
 |-  ( R e. V -> ( ( A e. dom QMap R /\ B e. dom QMap R ) <-> ( A e. dom R /\ B e. dom R ) ) )
7 6 pm5.32i 
 |-  ( ( R e. V /\ ( A e. dom QMap R /\ B e. dom QMap R ) ) <-> ( R e. V /\ ( A e. dom R /\ B e. dom R ) ) )
8 7 imbi1i 
 |-  ( ( ( R e. V /\ ( A e. dom QMap R /\ B e. dom QMap R ) ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) <-> ( ( R e. V /\ ( A e. dom R /\ B e. dom R ) ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) )
9 ecqmap 
 |-  ( A e. dom R -> [ A ] QMap R = { [ A ] R } )
10 ecqmap 
 |-  ( B e. dom R -> [ B ] QMap R = { [ B ] R } )
11 9 10 eqeqan12d 
 |-  ( ( A e. dom R /\ B e. dom R ) -> ( [ A ] QMap R = [ B ] QMap R <-> { [ A ] R } = { [ B ] R } ) )
12 11 imbi1d 
 |-  ( ( A e. dom R /\ B e. dom R ) -> ( ( [ A ] QMap R = [ B ] QMap R -> A = B ) <-> ( { [ A ] R } = { [ B ] R } -> A = B ) ) )
13 ecexg 
 |-  ( R e. V -> [ A ] R e. _V )
14 sneqbg 
 |-  ( [ A ] R e. _V -> ( { [ A ] R } = { [ B ] R } <-> [ A ] R = [ B ] R ) )
15 13 14 syl 
 |-  ( R e. V -> ( { [ A ] R } = { [ B ] R } <-> [ A ] R = [ B ] R ) )
16 15 imbi1d 
 |-  ( R e. V -> ( ( { [ A ] R } = { [ B ] R } -> A = B ) <-> ( [ A ] R = [ B ] R -> A = B ) ) )
17 12 16 sylan9bbr 
 |-  ( ( R e. V /\ ( A e. dom R /\ B e. dom R ) ) -> ( ( [ A ] QMap R = [ B ] QMap R -> A = B ) <-> ( [ A ] R = [ B ] R -> A = B ) ) )
18 17 pm5.74i 
 |-  ( ( ( R e. V /\ ( A e. dom R /\ B e. dom R ) ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) <-> ( ( R e. V /\ ( A e. dom R /\ B e. dom R ) ) -> ( [ A ] R = [ B ] R -> A = B ) ) )
19 8 18 bitri 
 |-  ( ( ( R e. V /\ ( A e. dom QMap R /\ B e. dom QMap R ) ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) <-> ( ( R e. V /\ ( A e. dom R /\ B e. dom R ) ) -> ( [ A ] R = [ B ] R -> A = B ) ) )
20 impexp 
 |-  ( ( ( R e. V /\ ( A e. dom QMap R /\ B e. dom QMap R ) ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) <-> ( R e. V -> ( ( A e. dom QMap R /\ B e. dom QMap R ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) ) )
21 impexp 
 |-  ( ( ( R e. V /\ ( A e. dom R /\ B e. dom R ) ) -> ( [ A ] R = [ B ] R -> A = B ) ) <-> ( R e. V -> ( ( A e. dom R /\ B e. dom R ) -> ( [ A ] R = [ B ] R -> A = B ) ) ) )
22 19 20 21 3bitr3i 
 |-  ( ( R e. V -> ( ( A e. dom QMap R /\ B e. dom QMap R ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) ) <-> ( R e. V -> ( ( A e. dom R /\ B e. dom R ) -> ( [ A ] R = [ B ] R -> A = B ) ) ) )
23 22 pm5.74ri 
 |-  ( R e. V -> ( ( ( A e. dom QMap R /\ B e. dom QMap R ) -> ( [ A ] QMap R = [ B ] QMap R -> A = B ) ) <-> ( ( A e. dom R /\ B e. dom R ) -> ( [ A ] R = [ B ] R -> A = B ) ) ) )
24 2 23 imbitrid 
 |-  ( R e. V -> ( Disj QMap R -> ( ( A e. dom R /\ B e. dom R ) -> ( [ A ] R = [ B ] R -> A = B ) ) ) )
25 1 24 sylbid 
 |-  ( R e. V -> ( QMap R e. Disjs -> ( ( A e. dom R /\ B e. dom R ) -> ( [ A ] R = [ B ] R -> A = B ) ) ) )
26 25 3imp 
 |-  ( ( R e. V /\ QMap R e. Disjs /\ ( A e. dom R /\ B e. dom R ) ) -> ( [ A ] R = [ B ] R -> A = B ) )