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Theorem hmphdis 22399
 Description: Homeomorphisms preserve topological discretion. (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 4850 . . . 4 𝐽 ⊆ 𝒫 𝐽
2 hmphdis.1 . . . . 5 𝑋 = 𝐽
32pweqi 4529 . . . 4 𝒫 𝑋 = 𝒫 𝐽
41, 3sseqtrri 3979 . . 3 𝐽 ⊆ 𝒫 𝑋
54a1i 11 . 2 (𝐽 ≃ 𝒫 𝐴𝐽 ⊆ 𝒫 𝑋)
6 hmph 22379 . . 3 (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7 n0 4282 . . . 4 ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8 elpwi 4520 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
9 imassrn 5918 . . . . . . . . . . 11 (𝑓𝑥) ⊆ ran 𝑓
10 unipw 5320 . . . . . . . . . . . . . . 15 𝒫 𝐴 = 𝐴
1110eqcomi 2831 . . . . . . . . . . . . . 14 𝐴 = 𝒫 𝐴
122, 11hmeof1o 22367 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋1-1-onto𝐴)
13 f1of 6597 . . . . . . . . . . . . 13 (𝑓:𝑋1-1-onto𝐴𝑓:𝑋𝐴)
14 frn 6500 . . . . . . . . . . . . 13 (𝑓:𝑋𝐴 → ran 𝑓𝐴)
1512, 13, 143syl 18 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓𝐴)
1615adantr 484 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → ran 𝑓𝐴)
179, 16sstrid 3953 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ⊆ 𝐴)
18 vex 3472 . . . . . . . . . . . 12 𝑓 ∈ V
1918imaex 7607 . . . . . . . . . . 11 (𝑓𝑥) ∈ V
2019elpw 4515 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴)
2117, 20sylibr 237 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
222hmeoopn 22369 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑥𝐽 ↔ (𝑓𝑥) ∈ 𝒫 𝐴))
2321, 22mpbird 260 . . . . . . . 8 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → 𝑥𝐽)
2423ex 416 . . . . . . 7 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥𝑋𝑥𝐽))
258, 24syl5 34 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋𝑥𝐽))
2625ssrdv 3948 . . . . 5 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
2726exlimiv 1931 . . . 4 (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
287, 27sylbi 220 . . 3 ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋𝐽)
296, 28sylbi 220 . 2 (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋𝐽)
305, 29eqssd 3959 1 (𝐽 ≃ 𝒫 𝐴𝐽 = 𝒫 𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2114   ≠ wne 3011   ⊆ wss 3908  ∅c0 4265  𝒫 cpw 4511  ∪ cuni 4813   class class class wbr 5042  ran crn 5533   “ cima 5535  ⟶wf 6330  –1-1-onto→wf1o 6333  (class class class)co 7140  Homeochmeo 22356   ≃ chmph 22357 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-1o 8089  df-map 8395  df-top 21497  df-topon 21514  df-cn 21830  df-hmeo 22358  df-hmph 22359 This theorem is referenced by: (None)
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