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Theorem qsss1 37095
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 4050 . . 3 (𝐴𝐵 → (∃𝑥𝐴 𝑦 = [𝑥]𝐶 → ∃𝑥𝐵 𝑦 = [𝑥]𝐶))
21ss2abdv 4059 . 2 (𝐴𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝐶} ⊆ {𝑦 ∣ ∃𝑥𝐵 𝑦 = [𝑥]𝐶})
3 df-qs 8705 . 2 (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝐶}
4 df-qs 8705 . 2 (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = [𝑥]𝐶}
52, 3, 43sstr4g 4026 1 (𝐴𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))
Colors of variables: wff setvar class
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)
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