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Theorem hmphdis 21879
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 4632 . . . 4 𝐽 ⊆ 𝒫 𝐽
2 hmphdis.1 . . . . 5 𝑋 = 𝐽
32pweqi 4319 . . . 4 𝒫 𝑋 = 𝒫 𝐽
41, 3sseqtr4i 3798 . . 3 𝐽 ⊆ 𝒫 𝑋
54a1i 11 . 2 (𝐽 ≃ 𝒫 𝐴𝐽 ⊆ 𝒫 𝑋)
6 hmph 21859 . . 3 (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7 n0 4095 . . . 4 ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8 elpwi 4325 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
9 imassrn 5659 . . . . . . . . . . 11 (𝑓𝑥) ⊆ ran 𝑓
10 unipw 5074 . . . . . . . . . . . . . . 15 𝒫 𝐴 = 𝐴
1110eqcomi 2774 . . . . . . . . . . . . . 14 𝐴 = 𝒫 𝐴
122, 11hmeof1o 21847 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋1-1-onto𝐴)
13 f1of 6320 . . . . . . . . . . . . 13 (𝑓:𝑋1-1-onto𝐴𝑓:𝑋𝐴)
14 frn 6229 . . . . . . . . . . . . 13 (𝑓:𝑋𝐴 → ran 𝑓𝐴)
1512, 13, 143syl 18 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓𝐴)
1615adantr 472 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → ran 𝑓𝐴)
179, 16syl5ss 3772 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ⊆ 𝐴)
18 vex 3353 . . . . . . . . . . . 12 𝑓 ∈ V
1918imaex 7302 . . . . . . . . . . 11 (𝑓𝑥) ∈ V
2019elpw 4321 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴)
2117, 20sylibr 225 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
222hmeoopn 21849 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑥𝐽 ↔ (𝑓𝑥) ∈ 𝒫 𝐴))
2321, 22mpbird 248 . . . . . . . 8 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → 𝑥𝐽)
2423ex 401 . . . . . . 7 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥𝑋𝑥𝐽))
258, 24syl5 34 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋𝑥𝐽))
2625ssrdv 3767 . . . . 5 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
2726exlimiv 2025 . . . 4 (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
287, 27sylbi 208 . . 3 ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋𝐽)
296, 28sylbi 208 . 2 (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋𝐽)
305, 29eqssd 3778 1 (𝐽 ≃ 𝒫 𝐴𝐽 = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wex 1874  wcel 2155  wne 2937  wss 3732  c0 4079  𝒫 cpw 4315   cuni 4594   class class class wbr 4809  ran crn 5278  cima 5280  wf 6064  1-1-ontowf1o 6067  (class class class)co 6842  Homeochmeo 21836  chmph 21837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-1o 7764  df-map 8062  df-top 20978  df-topon 20995  df-cn 21311  df-hmeo 21838  df-hmph 21839
This theorem is referenced by: (None)
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