Theorem qsss1 38972

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 4007	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶 → ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶))
2	1	ss2abdv 4019	. 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} ⊆ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶})
3		df-qs 8696	. 2 ⊢ (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶}
4		df-qs 8696	. 2 ⊢ (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶}
5	2, 3, 4	3sstr4g 3990	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  {cab 2741  ∃wrex 3089   ⊆ wss 3905  [cec 8688   / cqs 8689

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-rex 3090  df-ss 3922  df-qs 8696

This theorem is used by: (None)