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Theorem qsss1 38270
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 4064 . . 3 (𝐴𝐵 → (∃𝑥𝐴 𝑦 = [𝑥]𝐶 → ∃𝑥𝐵 𝑦 = [𝑥]𝐶))
21ss2abdv 4075 . 2 (𝐴𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝐶} ⊆ {𝑦 ∣ ∃𝑥𝐵 𝑦 = [𝑥]𝐶})
3 df-qs 8749 . 2 (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = [𝑥]𝐶}
4 df-qs 8749 . 2 (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥𝐵 𝑦 = [𝑥]𝐶}
52, 3, 43sstr4g 4040 1 (𝐴𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  {cab 2711  wrex 3067  wss 3962  [cec 8741   / cqs 8742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-rex 3068  df-ss 3979  df-qs 8749
This theorem is referenced by: (None)
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