Theorem qsss1 37095

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

Assertion

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

Proof of Theorem qsss1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrexv 4050	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶 → ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶))
2	1	ss2abdv 4059	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} ⊆ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶})
3		df-qs 8705	. 2 ⊢ (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶}
4		df-qs 8705	. 2 ⊢ (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶}
5	2, 3, 4	3sstr4g 4026	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))

Syntax hints:   → wi 4   = wceq 1542  {cab 2710  ∃wrex 3071   ⊆ wss 3947  [cec 8697   / cqs 8698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3072  df-v 3477  df-in 3954  df-ss 3964  df-qs 8705

This theorem is referenced by: (None)