Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4474 | . . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽 | |
2 | hmphdis.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | pweqi 4162 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽 |
4 | 1, 3 | sseqtr4i 3638 | . . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋 |
5 | 4 | a1i 11 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋) |
6 | hmph 21579 | . . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅) | |
7 | n0 3931 | . . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴)) | |
8 | elpwi 4168 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
9 | imassrn 5477 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓 | |
10 | unipw 4918 | . . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
11 | 10 | eqcomi 2631 | . . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
12 | 2, 11 | hmeof1o 21567 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋–1-1-onto→𝐴) |
13 | f1of 6137 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋⟶𝐴) | |
14 | frn 6053 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋⟶𝐴 → ran 𝑓 ⊆ 𝐴) | |
15 | 12, 13, 14 | 3syl 18 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓 ⊆ 𝐴) |
16 | 15 | adantr 481 | . . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → ran 𝑓 ⊆ 𝐴) |
17 | 9, 16 | syl5ss 3614 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴) |
18 | vex 3203 | . . . . . . . . . . . 12 ⊢ 𝑓 ∈ V | |
19 | 18 | imaex 7104 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V |
20 | 19 | elpw 4164 | . . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝑓 “ 𝑥) ⊆ 𝐴) |
21 | 17, 20 | sylibr 224 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ∈ 𝒫 𝐴) |
22 | 2 | hmeoopn 21569 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑓 “ 𝑥) ∈ 𝒫 𝐴)) |
23 | 21, 22 | mpbird 247 | . . . . . . . 8 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ 𝐽) |
24 | 23 | ex 450 | . . . . . . 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 3609 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
27 | 26 | exlimiv 1858 | . . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
28 | 7, 27 | sylbi 207 | . . 3 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋 ⊆ 𝐽) |
29 | 6, 28 | sylbi 207 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋 ⊆ 𝐽) |
30 | 5, 29 | eqssd 3620 | 1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 ∪ cuni 4436 class class class wbr 4653 ran crn 5115 “ cima 5117 ⟶wf 5884 –1-1-onto→wf1o 5887 (class class class)co 6650 Homeochmeo 21556 ≃ chmph 21557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-1o 7560 df-map 7859 df-top 20699 df-topon 20716 df-cn 21031 df-hmeo 21558 df-hmph 21559 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |