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Theorem qsss1 37620
Description: Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.)
Assertion
Ref Expression
qsss1 (𝐴𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))

Proof of Theorem qsss1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrexv 4051 . . 3 (𝐴𝐵 → (∃𝑥𝐴 𝑦 = [𝑥]𝐶 → ∃𝑥𝐵 𝑦 = [𝑥]𝐶))
21ss2abdv 4060 . 2 (𝐴𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝐶} ⊆ {𝑦 ∣ ∃𝑥𝐵 𝑦 = [𝑥]𝐶})
3 df-qs 8715 . 2 (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝐶}
4 df-qs 8715 . 2 (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = [𝑥]𝐶}
52, 3, 43sstr4g 4027 1 (𝐴𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  {cab 2708  wrex 3069  wss 3948  [cec 8707   / cqs 8708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-v 3475  df-in 3955  df-ss 3965  df-qs 8715
This theorem is referenced by: (None)
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