Theorem rnmptssrn

Description: Inclusion relation for two ranges expressed in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	rnmptssrn.b	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
	rnmptssrn.y	$$⊢\left(\phi \wedge x\in A\right)\to \exists y\in CB=D$$
Assertion	rnmptssrn	$$⊢\phi \to \mathrm{ran}\left(x\in A⟼B\right)\subseteq \mathrm{ran}\left(y\in C⟼D\right)$$

Proof

Step	Hyp	Ref	Expression
1		rnmptssrn.b	$$⊢\left(\phi \wedge x\in A\right)\to B\in V$$
2		rnmptssrn.y	$$⊢\left(\phi \wedge x\in A\right)\to \exists y\in CB=D$$
3		eqid	$$⊢\left(y\in C⟼D\right)=\left(y\in C⟼D\right)$$
4	3 2 1	elrnmptd	$$⊢\left(\phi \wedge x\in A\right)\to B\in \mathrm{ran}\left(y\in C⟼D\right)$$
5	4	ralrimiva	$$⊢\phi \to \forall x\in AB\in \mathrm{ran}\left(y\in C⟼D\right)$$
6		eqid	$$⊢\left(x\in A⟼B\right)=\left(x\in A⟼B\right)$$
7	6	rnmptss	$$⊢\forall x\in AB\in \mathrm{ran}\left(y\in C⟼D\right)\to \mathrm{ran}\left(x\in A⟼B\right)\subseteq \mathrm{ran}\left(y\in C⟼D\right)$$
8	5 7	syl	$$⊢\phi \to \mathrm{ran}\left(x\in A⟼B\right)\subseteq \mathrm{ran}\left(y\in C⟼D\right)$$