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Theorem hmphdis 23804
Description: Homeomorphisms preserve topological discreteness. (Contributed by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
hmphdis.1 𝑋 = 𝐽
Assertion
Ref Expression
hmphdis (𝐽 ≃ 𝒫 𝐴𝐽 = 𝒫 𝑋)

Proof of Theorem hmphdis
Dummy variables 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwuni 4945 . . . 4 𝐽 ⊆ 𝒫 𝐽
2 hmphdis.1 . . . . 5 𝑋 = 𝐽
32pweqi 4616 . . . 4 𝒫 𝑋 = 𝒫 𝐽
41, 3sseqtrri 4033 . . 3 𝐽 ⊆ 𝒫 𝑋
54a1i 11 . 2 (𝐽 ≃ 𝒫 𝐴𝐽 ⊆ 𝒫 𝑋)
6 hmph 23784 . . 3 (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7 n0 4353 . . . 4 ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8 elpwi 4607 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
9 imassrn 6089 . . . . . . . . . . 11 (𝑓𝑥) ⊆ ran 𝑓
10 unipw 5455 . . . . . . . . . . . . . . 15 𝒫 𝐴 = 𝐴
1110eqcomi 2746 . . . . . . . . . . . . . 14 𝐴 = 𝒫 𝐴
122, 11hmeof1o 23772 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋1-1-onto𝐴)
13 f1of 6848 . . . . . . . . . . . . 13 (𝑓:𝑋1-1-onto𝐴𝑓:𝑋𝐴)
14 frn 6743 . . . . . . . . . . . . 13 (𝑓:𝑋𝐴 → ran 𝑓𝐴)
1512, 13, 143syl 18 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓𝐴)
1615adantr 480 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → ran 𝑓𝐴)
179, 16sstrid 3995 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ⊆ 𝐴)
18 vex 3484 . . . . . . . . . . . 12 𝑓 ∈ V
1918imaex 7936 . . . . . . . . . . 11 (𝑓𝑥) ∈ V
2019elpw 4604 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴)
2117, 20sylibr 234 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
222hmeoopn 23774 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑥𝐽 ↔ (𝑓𝑥) ∈ 𝒫 𝐴))
2321, 22mpbird 257 . . . . . . . 8 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → 𝑥𝐽)
2423ex 412 . . . . . . 7 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥𝑋𝑥𝐽))
258, 24syl5 34 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋𝑥𝐽))
2625ssrdv 3989 . . . . 5 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
2726exlimiv 1930 . . . 4 (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
287, 27sylbi 217 . . 3 ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋𝐽)
296, 28sylbi 217 . 2 (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋𝐽)
305, 29eqssd 4001 1 (𝐽 ≃ 𝒫 𝐴𝐽 = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wne 2940  wss 3951  c0 4333  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  ran crn 5686  cima 5688  wf 6557  1-1-ontowf1o 6560  (class class class)co 7431  Homeochmeo 23761  chmph 23762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-1o 8506  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235  df-hmeo 23763  df-hmph 23764
This theorem is referenced by: (None)
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