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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 4037 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶 → ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶)) | |
2 | 1 | ss2abdv 4047 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} ⊆ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶}) |
3 | df-qs 8298 | . 2 ⊢ (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} | |
4 | df-qs 8298 | . 2 ⊢ (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶} | |
5 | 2, 3, 4 | 3sstr4g 4015 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 {cab 2802 ∃wrex 3142 ⊆ wss 3939 [cec 8290 / cqs 8291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-rex 3147 df-in 3946 df-ss 3955 df-qs 8298 |
This theorem is referenced by: (None) |
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