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Mirrors > Home > MPE Home > Th. List > Mathboxes > fneint | Structured version Visualization version GIF version |
Description: If a cover is finer than another, every point can be approached more closely by intersections. (Contributed by Jeff Hankins, 11-Oct-2009.) |
Ref | Expression |
---|---|
fneint | ⊢ (𝐴Fne𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2w 2893 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦)) | |
2 | 1 | elrab 3677 | . . . 4 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) |
3 | fnessex 33591 | . . . . . . 7 ⊢ ((𝐴Fne𝐵 ∧ 𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) | |
4 | 3 | 3expb 1112 | . . . . . 6 ⊢ ((𝐴Fne𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) → ∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦)) |
5 | eleq2w 2893 | . . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧)) | |
6 | 5 | intminss 4893 | . . . . . . . . 9 ⊢ ((𝑧 ∈ 𝐵 ∧ 𝑃 ∈ 𝑧) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧) |
7 | sstr 3972 | . . . . . . . . 9 ⊢ ((∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦) | |
8 | 6, 7 | sylan 580 | . . . . . . . 8 ⊢ (((𝑧 ∈ 𝐵 ∧ 𝑃 ∈ 𝑧) ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦) |
9 | 8 | expl 458 | . . . . . . 7 ⊢ (𝑧 ∈ 𝐵 → ((𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)) |
10 | 9 | rexlimiv 3277 | . . . . . 6 ⊢ (∃𝑧 ∈ 𝐵 (𝑃 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦) |
11 | 4, 10 | syl 17 | . . . . 5 ⊢ ((𝐴Fne𝐵 ∧ (𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦)) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦) |
12 | 11 | ex 413 | . . . 4 ⊢ (𝐴Fne𝐵 → ((𝑦 ∈ 𝐴 ∧ 𝑃 ∈ 𝑦) → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)) |
13 | 2, 12 | syl5bi 243 | . . 3 ⊢ (𝐴Fne𝐵 → (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦)) |
14 | 13 | ralrimiv 3178 | . 2 ⊢ (𝐴Fne𝐵 → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦) |
15 | ssint 4883 | . 2 ⊢ (∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ 𝑦) | |
16 | 14, 15 | sylibr 235 | 1 ⊢ (𝐴Fne𝐵 → ∩ {𝑥 ∈ 𝐵 ∣ 𝑃 ∈ 𝑥} ⊆ ∩ {𝑥 ∈ 𝐴 ∣ 𝑃 ∈ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 {crab 3139 ⊆ wss 3933 ∩ cint 4867 class class class wbr 5057 Fnecfne 33581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-int 4868 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-topgen 16705 df-fne 33582 |
This theorem is referenced by: (None) |
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