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	Could not format assertion : No typesetting found for |- ( ( 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 ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		rnqmap	Could not format ran QMap R = ( dom R /. R ) : No typesetting found for |- ran QMap R = ( dom R /. R ) with typecode |-
2	1	eleq1i	Could not format ( ran QMap R e. ElDisjs <-> ( dom R /. R ) e. ElDisjs ) : No typesetting found for |- ( ran QMap R e. ElDisjs <-> ( dom R /. R ) e. ElDisjs ) with typecode |-
3		dmqsex	$$⊢R\in V\to \mathrm{dom}R/R\in \mathrm{V}$$
4		eleldisjseldisj	$$⊢\mathrm{dom}R/R\in \mathrm{V}\to \left(\mathrm{dom}R/R\in \mathrm{ElDisjs}\leftrightarrow ElDisj\left(\mathrm{dom}R/R\right)\right)$$
5	3 4	syl	$$⊢R\in V\to \left(\mathrm{dom}R/R\in \mathrm{ElDisjs}\leftrightarrow ElDisj\left(\mathrm{dom}R/R\right)\right)$$
6	2 5	bitrid	Could not format ( R e. V -> ( ran QMap R e. ElDisjs <-> ElDisj ( dom R /. R ) ) ) : No typesetting found for |- ( R e. V -> ( ran QMap R e. ElDisjs <-> ElDisj ( dom R /. R ) ) ) with typecode |-
7		eldisjim3	$$⊢ElDisj\left(\mathrm{dom}R/R\right)\to \left(\left(\left[A\right]R\in \left(\mathrm{dom}R/R\right)\wedge \left[B\right]R\in \left(\mathrm{dom}R/R\right)\right)\to \left(\left[A\right]R\cap \left[B\right]R\ne \varnothing \to \left[A\right]R=\left[B\right]R\right)\right)$$
8		eceldmqs	$$⊢R\in V\to \left(\left[A\right]R\in \left(\mathrm{dom}R/R\right)\leftrightarrow A\in \mathrm{dom}R\right)$$
9		eceldmqs	$$⊢R\in V\to \left(\left[B\right]R\in \left(\mathrm{dom}R/R\right)\leftrightarrow B\in \mathrm{dom}R\right)$$
10	8 9	anbi12d	$$⊢R\in V\to \left(\left(\left[A\right]R\in \left(\mathrm{dom}R/R\right)\wedge \left[B\right]R\in \left(\mathrm{dom}R/R\right)\right)\leftrightarrow \left(A\in \mathrm{dom}R\wedge B\in \mathrm{dom}R\right)\right)$$
11	10	imbi1d	$$⊢R\in V\to \left(\left(\left(\left[A\right]R\in \left(\mathrm{dom}R/R\right)\wedge \left[B\right]R\in \left(\mathrm{dom}R/R\right)\right)\to \left(\left[A\right]R\cap \left[B\right]R\ne \varnothing \to \left[A\right]R=\left[B\right]R\right)\right)\leftrightarrow \left(\left(A\in \mathrm{dom}R\wedge B\in \mathrm{dom}R\right)\to \left(\left[A\right]R\cap \left[B\right]R\ne \varnothing \to \left[A\right]R=\left[B\right]R\right)\right)\right)$$
12	7 11	imbitrid	$$⊢R\in V\to \left(ElDisj\left(\mathrm{dom}R/R\right)\to \left(\left(A\in \mathrm{dom}R\wedge B\in \mathrm{dom}R\right)\to \left(\left[A\right]R\cap \left[B\right]R\ne \varnothing \to \left[A\right]R=\left[B\right]R\right)\right)\right)$$
13	6 12	sylbid	Could not format ( 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 ) ) ) ) : No typesetting found for |- ( 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 ) ) ) ) with typecode |-
14	13	3imp	Could not format ( ( 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 ) ) : No typesetting found for |- ( ( 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 ) ) with typecode |-