rnmptssrn

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

	Ref	Expression
Hypotheses	rnmptssrn.b	$⊢ (φ \land x \in A) \to B \in V$
	rnmptssrn.y	$⊢ (φ \land x \in A) \to \exists y \in C B = D$
Assertion	rnmptssrn	$⊢ φ \to ran (x \in A ⟼ B) \subseteq ran (y \in C ⟼ D)$

Step	Hyp	Ref	Expression
1		rnmptssrn.b	$⊢ (φ \land x \in A) \to B \in V$
2		rnmptssrn.y	$⊢ (φ \land x \in A) \to \exists y \in C B = D$
3		eqid	$⊢ (y \in C ⟼ D) = (y \in C ⟼ D)$
4	3 2 1	elrnmptd	$⊢ (φ \land x \in A) \to B \in ran (y \in C ⟼ D)$
5	4	ralrimiva	$⊢ φ \to \forall x \in A B \in ran (y \in C ⟼ D)$
6		eqid	$⊢ (x \in A ⟼ B) = (x \in A ⟼ B)$
7	6	rnmptss	$⊢ \forall x \in A B \in ran (y \in C ⟼ D) \to ran (x \in A ⟼ B) \subseteq ran (y \in C ⟼ D)$
8	5 7	syl	$⊢ φ \to ran (x \in A ⟼ B) \subseteq ran (y \in C ⟼ D)$

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