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Metamath Proof Explorer

Theorem disjimrmoeqec

Description: Under Disj , every block has a unique generator ( E* form). If t is a block in the quotient sense, then there is a uniquely determined u in dom R such that t = [ u ] R . This is the existence+uniqueness engine behind Disjs and QMap characterizations: it is the "representative theorem" from which the E! forms are obtained. (Contributed by Peter Mazsa, 5-Feb-2026)

Ref Expression
Assertion disjimrmoeqec 
|- ( Disj R -> E* u e. dom R t = [ u ] R )

Proof

Step Hyp Ref Expression
1 disjimeceqim 
 |-  ( Disj R -> A. u e. dom R A. v e. dom R ( [ u ] R = [ v ] R -> u = v ) )
2 eqtr2 
 |-  ( ( t = [ u ] R /\ t = [ v ] R ) -> [ u ] R = [ v ] R )
3 2 imim1i 
 |-  ( ( [ u ] R = [ v ] R -> u = v ) -> ( ( t = [ u ] R /\ t = [ v ] R ) -> u = v ) )
4 3 2ralimi 
 |-  ( A. u e. dom R A. v e. dom R ( [ u ] R = [ v ] R -> u = v ) -> A. u e. dom R A. v e. dom R ( ( t = [ u ] R /\ t = [ v ] R ) -> u = v ) )
5 1 4 syl 
 |-  ( Disj R -> A. u e. dom R A. v e. dom R ( ( t = [ u ] R /\ t = [ v ] R ) -> u = v ) )
6 eceq1 
 |-  ( u = v -> [ u ] R = [ v ] R )
7 6 eqeq2d 
 |-  ( u = v -> ( t = [ u ] R <-> t = [ v ] R ) )
8 7 rmo4 
 |-  ( E* u e. dom R t = [ u ] R <-> A. u e. dom R A. v e. dom R ( ( t = [ u ] R /\ t = [ v ] R ) -> u = v ) )
9 5 8 sylibr 
 |-  ( Disj R -> E* u e. dom R t = [ u ] R )