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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 4825 | . . . 4 ⊢ 𝐽 ⊆ 𝒫 ∪ 𝐽 | |
2 | hmphdis.1 | . . . . 5 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | pweqi 4112 | . . . 4 ⊢ 𝒫 𝑋 = 𝒫 ∪ 𝐽 |
4 | 1, 3 | sseqtr4i 3601 | . . 3 ⊢ 𝐽 ⊆ 𝒫 𝑋 |
5 | 4 | a1i 11 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 ⊆ 𝒫 𝑋) |
6 | hmph 21389 | . . 3 ⊢ (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅) | |
7 | n0 3890 | . . . 4 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴)) | |
8 | elpwi 4117 | . . . . . . 7 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋) | |
9 | imassrn 5396 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ⊆ ran 𝑓 | |
10 | unipw 4845 | . . . . . . . . . . . . . . 15 ⊢ ∪ 𝒫 𝐴 = 𝐴 | |
11 | 10 | eqcomi 2619 | . . . . . . . . . . . . . 14 ⊢ 𝐴 = ∪ 𝒫 𝐴 |
12 | 2, 11 | hmeof1o 21377 | . . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋–1-1-onto→𝐴) |
13 | f1of 6050 | . . . . . . . . . . . . 13 ⊢ (𝑓:𝑋–1-1-onto→𝐴 → 𝑓:𝑋⟶𝐴) | |
14 | frn 5966 | . . . . . . . . . . . . 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 | syl5ss 3579 | . . . . . . . . . 10 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ⊆ 𝐴) |
18 | vex 3176 | . . . . . . . . . . . 12 ⊢ 𝑓 ∈ V | |
19 | 18 | imaex 6996 | . . . . . . . . . . 11 ⊢ (𝑓 “ 𝑥) ∈ V |
20 | 19 | elpw 4114 | . . . . . . . . . 10 ⊢ ((𝑓 “ 𝑥) ∈ 𝒫 𝐴 ↔ (𝑓 “ 𝑥) ⊆ 𝐴) |
21 | 17, 20 | sylibr 223 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑓 “ 𝑥) ∈ 𝒫 𝐴) |
22 | 2 | hmeoopn 21379 | . . . . . . . . 9 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → (𝑥 ∈ 𝐽 ↔ (𝑓 “ 𝑥) ∈ 𝒫 𝐴)) |
23 | 21, 22 | mpbird 246 | . . . . . . . 8 ⊢ ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ 𝐽) |
24 | 23 | ex 449 | . . . . . . 7 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ⊆ 𝑋 → 𝑥 ∈ 𝐽)) |
25 | 8, 24 | syl5 33 | . . . . . 6 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋 → 𝑥 ∈ 𝐽)) |
26 | 25 | ssrdv 3574 | . . . . 5 ⊢ (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
27 | 26 | exlimiv 1845 | . . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋 ⊆ 𝐽) |
28 | 7, 27 | sylbi 206 | . . 3 ⊢ ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋 ⊆ 𝐽) |
29 | 6, 28 | sylbi 206 | . 2 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋 ⊆ 𝐽) |
30 | 5, 29 | eqssd 3585 | 1 ⊢ (𝐽 ≃ 𝒫 𝐴 → 𝐽 = 𝒫 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 ran crn 5039 “ cima 5041 ⟶wf 5800 –1-1-onto→wf1o 5803 (class class class)co 6549 Homeochmeo 21366 ≃ chmph 21367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-1o 7447 df-map 7746 df-top 20521 df-topon 20523 df-cn 20841 df-hmeo 21368 df-hmph 21369 |
This theorem is referenced by: (None) |
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