This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem qsss1

Description: Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020)

		Ref	Expression
	Assertion	qsss1	$⊢ A \subseteq B \to A / C \subseteq B / C$

Step	Hyp	Ref	Expression
1		ssrexv	$⊢ A \subseteq B \to (\exists x \in A y = [x] C \to \exists x \in B y = [x] C)$
2	1	ss2abdv	$⊢ A \subseteq B \to \{y \| \exists x \in A y = [x] C\} \subseteq \{y \| \exists x \in B y = [x] C\}$
3		df-qs	$⊢ A / C = \{y \| \exists x \in A y = [x] C\}$
4		df-qs	$⊢ B / C = \{y \| \exists x \in B y = [x] C\}$
5	2 3 4	3sstr4g	$⊢ A \subseteq B \to A / C \subseteq B / C$