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Mirrors > Home > MPE Home > Th. List > Mathboxes > qsss1 | Structured version Visualization version GIF version |
Description: Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.) |
Ref | Expression |
---|---|
qsss1 | ⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrexv 4033 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶 → ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶)) | |
2 | 1 | ss2abdv 4043 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} ⊆ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶}) |
3 | df-qs 8294 | . 2 ⊢ (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} | |
4 | df-qs 8294 | . 2 ⊢ (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶} | |
5 | 2, 3, 4 | 3sstr4g 4011 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {cab 2799 ∃wrex 3139 ⊆ wss 3935 [cec 8286 / cqs 8287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-in 3942 df-ss 3951 df-qs 8294 |
This theorem is referenced by: (None) |
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