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 )