Metamath Proof Explorer

Theorem disjqmap

Description: Disjointness of QMap equals unique generation of the quotient carrier. The cleaned, carrier-respecting version of disjqmap2 . This is the statement "each equivalence class has a unique representative" for the general coset carrier ( dom R /. R ) . (Contributed by Peter Mazsa, 12-Feb-2026)

		Ref	Expression
	Assertion	disjqmap	\|- ( R e. V -> ( Disj QMap R <-> A. u e. ( dom R /. R ) E! t e. dom R u = [ t ] R ) )

Proof

Step	Hyp	Ref	Expression
1		disjqmap2	\|- ( R e. V -> ( Disj QMap R <-> A. u E* t e. dom R u = [ t ] R ) )
2		raldmqseu	\|- ( R e. V -> ( A. u e. ( dom R /. R ) E! t e. dom R u = [ t ] R <-> A. u E* t e. dom R u = [ t ] R ) )
3	1 2	bitr4d	\|- ( R e. V -> ( Disj QMap R <-> A. u e. ( dom R /. R ) E! t e. dom R u = [ t ] R ) )

Step

Hyp

Ref

Expression

disjqmap2

 |-  ( R e. V -> ( Disj QMap R <-> A. u E* t e. dom R u = [ t ] R ) )

raldmqseu

 |-  ( R e. V -> ( A. u e. ( dom R /. R ) E! t e. dom R u = [ t ] R <-> A. u E* t e. dom R u = [ t ] R ) )

1 2

bitr4d

 |-  ( R e. V -> ( Disj QMap R <-> A. u e. ( dom R /. R ) E! t e. dom R u = [ t ] R ) )