Theorem qsss1

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

	Ref	Expression
Assertion	qsss1	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 / 𝐶 ) ⊆ ( 𝐵 / 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		ssrexv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 = [ 𝑥 ] 𝐶 → ∃ 𝑥 ∈ 𝐵 𝑦 = [ 𝑥 ] 𝐶 ) )
2	1	ss2abdv	⊢ ( 𝐴 ⊆ 𝐵 → { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = [ 𝑥 ] 𝐶 } ⊆ { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 = [ 𝑥 ] 𝐶 } )
3		df-qs	⊢ ( 𝐴 / 𝐶 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = [ 𝑥 ] 𝐶 }
4		df-qs	⊢ ( 𝐵 / 𝐶 ) = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐵 𝑦 = [ 𝑥 ] 𝐶 }
5	2 3 4	3sstr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 / 𝐶 ) ⊆ ( 𝐵 / 𝐶 ) )