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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 4946 | . . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽 | |
2 | hmphdis.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | pweqi 4614 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽 |
4 | 1, 3 | sseqtrri 4017 | . . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋 |
5 | 4 | a1i 11 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋) |
6 | hmph 23766 | . . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅) | |
7 | n0 4347 | . . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴)) | |
8 | elpwi 4605 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
9 | imassrn 6071 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓 | |
10 | unipw 5447 | . . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
11 | 10 | eqcomi 2735 | . . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
12 | 2, 11 | hmeof1o 23754 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋–1-1-onto→𝐴) |
13 | f1of 6833 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋⟶𝐴) | |
14 | frn 6725 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋⟶𝐴 → ran 𝑓 ⊆ 𝐴) | |
15 | 12, 13, 14 | 3syl 18 | . . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓 ⊆ 𝐴) |
16 | 15 | adantr 479 | . . . . . . . . . . 11 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → ran 𝑓 ⊆ 𝐴) |
17 | 9, 16 | sstrid 3991 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴) |
18 | vex 3467 | . . . . . . . . . . . 12 ⊢ 𝑓 ∈ V | |
19 | 18 | imaex 7917 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V |
20 | 19 | elpw 4602 | . . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝑓 “ 𝑥) ⊆ 𝐴) |
21 | 17, 20 | sylibr 233 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ∈ 𝒫 𝐴) |
22 | 2 | hmeoopn 23756 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑓 “ 𝑥) ∈ 𝒫 𝐴)) |
23 | 21, 22 | mpbird 256 | . . . . . . . 8 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ 𝐽) |
24 | 23 | ex 411 | . . . . . . 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 3985 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
27 | 26 | exlimiv 1926 | . . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
28 | 7, 27 | sylbi 216 | . . 3 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋 ⊆ 𝐽) |
29 | 6, 28 | sylbi 216 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋 ⊆ 𝐽) |
30 | 5, 29 | eqssd 3997 | 1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2930 ⊆ wss 3947 ∅c0 4323 𝒫 cpw 4598 ∪ cuni 4906 class class class wbr 5144 ran crn 5674 “ cima 5676 ⟶wf 6540 –1-1-onto→wf1o 6543 (class class class)co 7414 Homeochmeo 23743 ≃ chmph 23744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7993 df-2nd 7994 df-1o 8486 df-map 8847 df-top 22882 df-topon 22899 df-cn 23217 df-hmeo 23745 df-hmph 23746 |
This theorem is referenced by: (None) |
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