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Mirrors > Home > MPE Home > Th. List > Mathboxes > cossss | Structured version Visualization version GIF version |
Description: Subclass theorem for the classes of cosets by 𝐴 and 𝐵. (Contributed by Peter Mazsa, 11-Nov-2019.) |
Ref | Expression |
---|---|
cossss | ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssbr 5192 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑦 → 𝑥𝐵𝑦)) | |
2 | ssbr 5192 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → (𝑥𝐴𝑧 → 𝑥𝐵𝑧)) | |
3 | 1, 2 | anim12d 608 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → ((𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → (𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧))) |
4 | 3 | eximdv 1919 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧) → ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧))) |
5 | 4 | ssopab2dv 5551 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} ⊆ {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)}) |
6 | df-coss 37745 | . 2 ⊢ ≀ 𝐴 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐴𝑦 ∧ 𝑥𝐴𝑧)} | |
7 | df-coss 37745 | . 2 ⊢ ≀ 𝐵 = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑥𝐵𝑦 ∧ 𝑥𝐵𝑧)} | |
8 | 5, 6, 7 | 3sstr4g 4027 | 1 ⊢ (𝐴 ⊆ 𝐵 → ≀ 𝐴 ⊆ ≀ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1780 ⊆ wss 3948 class class class wbr 5148 {copab 5210 ≀ ccoss 37507 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-in 3955 df-ss 3965 df-br 5149 df-opab 5211 df-coss 37745 |
This theorem is referenced by: funALTVss 38033 |
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