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Theorem hmphdis 22098
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 4742 . . . 4 𝐽 ⊆ 𝒫 𝐽
2 hmphdis.1 . . . . 5 𝑋 = 𝐽
32pweqi 4420 . . . 4 𝒫 𝑋 = 𝒫 𝐽
41, 3sseqtr4i 3890 . . 3 𝐽 ⊆ 𝒫 𝑋
54a1i 11 . 2 (𝐽 ≃ 𝒫 𝐴𝐽 ⊆ 𝒫 𝑋)
6 hmph 22078 . . 3 (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7 n0 4191 . . . 4 ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8 elpwi 4426 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
9 imassrn 5775 . . . . . . . . . . 11 (𝑓𝑥) ⊆ ran 𝑓
10 unipw 5192 . . . . . . . . . . . . . . 15 𝒫 𝐴 = 𝐴
1110eqcomi 2781 . . . . . . . . . . . . . 14 𝐴 = 𝒫 𝐴
122, 11hmeof1o 22066 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋1-1-onto𝐴)
13 f1of 6438 . . . . . . . . . . . . 13 (𝑓:𝑋1-1-onto𝐴𝑓:𝑋𝐴)
14 frn 6344 . . . . . . . . . . . . 13 (𝑓:𝑋𝐴 → ran 𝑓𝐴)
1512, 13, 143syl 18 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓𝐴)
1615adantr 473 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → ran 𝑓𝐴)
179, 16syl5ss 3865 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ⊆ 𝐴)
18 vex 3412 . . . . . . . . . . . 12 𝑓 ∈ V
1918imaex 7430 . . . . . . . . . . 11 (𝑓𝑥) ∈ V
2019elpw 4422 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴)
2117, 20sylibr 226 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
222hmeoopn 22068 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑥𝐽 ↔ (𝑓𝑥) ∈ 𝒫 𝐴))
2321, 22mpbird 249 . . . . . . . 8 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → 𝑥𝐽)
2423ex 405 . . . . . . 7 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥𝑋𝑥𝐽))
258, 24syl5 34 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋𝑥𝐽))
2625ssrdv 3860 . . . . 5 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
2726exlimiv 1889 . . . 4 (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
287, 27sylbi 209 . . 3 ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋𝐽)
296, 28sylbi 209 . 2 (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋𝐽)
305, 29eqssd 3871 1 (𝐽 ≃ 𝒫 𝐴𝐽 = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wex 1742  wcel 2048  wne 2961  wss 3825  c0 4173  𝒫 cpw 4416   cuni 4706   class class class wbr 4923  ran crn 5401  cima 5403  wf 6178  1-1-ontowf1o 6181  (class class class)co 6970  Homeochmeo 22055  chmph 22056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-1o 7897  df-map 8200  df-top 21196  df-topon 21213  df-cn 21529  df-hmeo 22057  df-hmph 22058
This theorem is referenced by: (None)
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