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Mirrors > Home > MPE Home > Th. List > Mathboxes > crefss | Structured version Visualization version GIF version |
Description: The "every open cover has an 𝐴 refinement" predicate respects inclusion. (Contributed by Thierry Arnoux, 7-Jan-2020.) |
Ref | Expression |
---|---|
crefss | ⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sslin 4210 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → (𝒫 𝑗 ∩ 𝐴) ⊆ (𝒫 𝑗 ∩ 𝐵)) | |
2 | ssrexv 4033 | . . . . . . 7 ⊢ ((𝒫 𝑗 ∩ 𝐴) ⊆ (𝒫 𝑗 ∩ 𝐵) → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)) | |
3 | 1, 2 | syl 17 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)) |
4 | 3 | imim2d 57 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → ((∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) → (∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦))) |
5 | 4 | ralimdv 3178 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦) → ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦))) |
6 | 5 | anim2d 613 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ((𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦)) → (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦)))) |
7 | eqid 2821 | . . . 4 ⊢ ∪ 𝑗 = ∪ 𝑗 | |
8 | 7 | iscref 31103 | . . 3 ⊢ (𝑗 ∈ CovHasRef𝐴 ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐴)𝑧Ref𝑦))) |
9 | 7 | iscref 31103 | . . 3 ⊢ (𝑗 ∈ CovHasRef𝐵 ↔ (𝑗 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝑗(∪ 𝑗 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑗 ∩ 𝐵)𝑧Ref𝑦))) |
10 | 6, 8, 9 | 3imtr4g 298 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑗 ∈ CovHasRef𝐴 → 𝑗 ∈ CovHasRef𝐵)) |
11 | 10 | ssrdv 3972 | 1 ⊢ (𝐴 ⊆ 𝐵 → CovHasRef𝐴 ⊆ CovHasRef𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4538 ∪ cuni 4831 class class class wbr 5058 Topctop 21495 Refcref 22104 CovHasRefccref 31101 |
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-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-in 3942 df-ss 3951 df-pw 4540 df-uni 4832 df-cref 31102 |
This theorem is referenced by: (None) |
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