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Mirrors > Home > MPE Home > Th. List > hmphdis | Structured version Visualization version GIF version |
Description: Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
hmphdis.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
hmphdis | ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwuni 4874 | . . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽 | |
2 | hmphdis.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | pweqi 4556 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽 |
4 | 1, 3 | sseqtrri 4003 | . . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋 |
5 | 4 | a1i 11 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋) |
6 | hmph 22383 | . . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅) | |
7 | n0 4309 | . . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴)) | |
8 | elpwi 4547 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
9 | imassrn 5939 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓 | |
10 | unipw 5342 | . . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
11 | 10 | eqcomi 2830 | . . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
12 | 2, 11 | hmeof1o 22371 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋–1-1-onto→𝐴) |
13 | f1of 6614 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋⟶𝐴) | |
14 | frn 6519 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋⟶𝐴 → ran 𝑓 ⊆ 𝐴) | |
15 | 12, 13, 14 | 3syl 18 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓 ⊆ 𝐴) |
16 | 15 | adantr 483 | . . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → ran 𝑓 ⊆ 𝐴) |
17 | 9, 16 | sstrid 3977 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴) |
18 | vex 3497 | . . . . . . . . . . . 12 ⊢ 𝑓 ∈ V | |
19 | 18 | imaex 7620 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V |
20 | 19 | elpw 4542 | . . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝑓 “ 𝑥) ⊆ 𝐴) |
21 | 17, 20 | sylibr 236 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ∈ 𝒫 𝐴) |
22 | 2 | hmeoopn 22373 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑓 “ 𝑥) ∈ 𝒫 𝐴)) |
23 | 21, 22 | mpbird 259 | . . . . . . . 8 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ 𝐽) |
24 | 23 | ex 415 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ⊆ 𝑋 → 𝑥 ∈ 𝐽)) |
25 | 8, 24 | syl5 34 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋 → 𝑥 ∈ 𝐽)) |
26 | 25 | ssrdv 3972 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
27 | 26 | exlimiv 1927 | . . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
28 | 7, 27 | sylbi 219 | . . 3 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋 ⊆ 𝐽) |
29 | 6, 28 | sylbi 219 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋 ⊆ 𝐽) |
30 | 5, 29 | eqssd 3983 | 1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4837 class class class wbr 5065 ran crn 5555 “ cima 5557 ⟶wf 6350 –1-1-onto→wf1o 6353 (class class class)co 7155 Homeochmeo 22360 ≃ chmph 22361 |
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 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-1o 8101 df-map 8407 df-top 21501 df-topon 21518 df-cn 21834 df-hmeo 22362 df-hmph 22363 |
This theorem is referenced by: (None) |
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