![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hmphdis | Structured version Visualization version GIF version |
Description: Homeomorphisms preserve topological discreteness. (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 4950 | . . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽 | |
2 | hmphdis.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | pweqi 4621 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽 |
4 | 1, 3 | sseqtrri 4033 | . . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋 |
5 | 4 | a1i 11 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋) |
6 | hmph 23800 | . . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅) | |
7 | n0 4359 | . . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴)) | |
8 | elpwi 4612 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
9 | imassrn 6091 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓 | |
10 | unipw 5461 | . . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
11 | 10 | eqcomi 2744 | . . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
12 | 2, 11 | hmeof1o 23788 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋–1-1-onto→𝐴) |
13 | f1of 6849 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋⟶𝐴) | |
14 | frn 6744 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋⟶𝐴 → ran 𝑓 ⊆ 𝐴) | |
15 | 12, 13, 14 | 3syl 18 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓 ⊆ 𝐴) |
16 | 15 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → ran 𝑓 ⊆ 𝐴) |
17 | 9, 16 | sstrid 4007 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴) |
18 | vex 3482 | . . . . . . . . . . . 12 ⊢ 𝑓 ∈ V | |
19 | 18 | imaex 7937 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V |
20 | 19 | elpw 4609 | . . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝑓 “ 𝑥) ⊆ 𝐴) |
21 | 17, 20 | sylibr 234 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ∈ 𝒫 𝐴) |
22 | 2 | hmeoopn 23790 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑓 “ 𝑥) ∈ 𝒫 𝐴)) |
23 | 21, 22 | mpbird 257 | . . . . . . . 8 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ 𝐽) |
24 | 23 | ex 412 | . . . . . . 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 4001 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
27 | 26 | exlimiv 1928 | . . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
28 | 7, 27 | sylbi 217 | . . 3 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋 ⊆ 𝐽) |
29 | 6, 28 | sylbi 217 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋 ⊆ 𝐽) |
30 | 5, 29 | eqssd 4013 | 1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 class class class wbr 5148 ran crn 5690 “ cima 5692 ⟶wf 6559 –1-1-onto→wf1o 6562 (class class class)co 7431 Homeochmeo 23777 ≃ chmph 23778 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-map 8867 df-top 22916 df-topon 22933 df-cn 23251 df-hmeo 23779 df-hmph 23780 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |