Theorem metreslem

Description: Lemma for metres . (Contributed by Mario Carneiro, 24-Aug-2015)

	Ref	Expression
Assertion	metreslem	$$⊢\mathrm{dom}D=X\times X\to {D\upharpoonright }_{\left(R\times R\right)}={D\upharpoonright }_{\left(\left(X\cap R\right)\times \left(X\cap R\right)\right)}$$

Proof

Step	Hyp	Ref	Expression
1		resdmres	$$⊢{D\upharpoonright }_{\mathrm{dom}\left({D\upharpoonright }_{\left(R\times R\right)}\right)}={D\upharpoonright }_{\left(R\times R\right)}$$
2		ineq2	$$⊢\mathrm{dom}D=X\times X\to \left(R\times R\right)\cap \mathrm{dom}D=\left(R\times R\right)\cap \left(X\times X\right)$$
3		dmres	$$⊢\mathrm{dom}\left({D\upharpoonright }_{\left(R\times R\right)}\right)=\left(R\times R\right)\cap \mathrm{dom}D$$
4		inxp	$$⊢\left(X\times X\right)\cap \left(R\times R\right)=\left(X\cap R\right)\times \left(X\cap R\right)$$
5		incom	$$⊢\left(X\times X\right)\cap \left(R\times R\right)=\left(R\times R\right)\cap \left(X\times X\right)$$
6	4 5	eqtr3i	$$⊢\left(X\cap R\right)\times \left(X\cap R\right)=\left(R\times R\right)\cap \left(X\times X\right)$$
7	2 3 6	3eqtr4g	$$⊢\mathrm{dom}D=X\times X\to \mathrm{dom}\left({D\upharpoonright }_{\left(R\times R\right)}\right)=\left(X\cap R\right)\times \left(X\cap R\right)$$
8	7	reseq2d	$$⊢\mathrm{dom}D=X\times X\to {D\upharpoonright }_{\mathrm{dom}\left({D\upharpoonright }_{\left(R\times R\right)}\right)}={D\upharpoonright }_{\left(\left(X\cap R\right)\times \left(X\cap R\right)\right)}$$
9	1 8	syl5eqr	$$⊢\mathrm{dom}D=X\times X\to {D\upharpoonright }_{\left(R\times R\right)}={D\upharpoonright }_{\left(\left(X\cap R\right)\times \left(X\cap R\right)\right)}$$