| Mathbox for Peter Mazsa |
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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 4053 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶 → ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶)) | |
| 2 | 1 | ss2abdv 4066 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} ⊆ {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶}) |
| 3 | df-qs 8751 | . 2 ⊢ (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} | |
| 4 | df-qs 8751 | . 2 ⊢ (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶} | |
| 5 | 2, 3, 4 | 3sstr4g 4037 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 / 𝐶) ⊆ (𝐵 / 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 {cab 2714 ∃wrex 3070 ⊆ wss 3951 [cec 8743 / cqs 8744 |
| 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 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-rex 3071 df-ss 3968 df-qs 8751 |
| This theorem is referenced by: (None) |
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