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Theorem hmphdis 21522
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 4864 . . . 4 𝐽 ⊆ 𝒫 𝐽
2 hmphdis.1 . . . . 5 𝑋 = 𝐽
32pweqi 4139 . . . 4 𝒫 𝑋 = 𝒫 𝐽
41, 3sseqtr4i 3622 . . 3 𝐽 ⊆ 𝒫 𝑋
54a1i 11 . 2 (𝐽 ≃ 𝒫 𝐴𝐽 ⊆ 𝒫 𝑋)
6 hmph 21502 . . 3 (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) ≠ ∅)
7 n0 3912 . . . 4 ((𝐽Homeo𝒫 𝐴) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8 elpwi 4145 . . . . . . 7 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
9 imassrn 5441 . . . . . . . . . . 11 (𝑓𝑥) ⊆ ran 𝑓
10 unipw 4884 . . . . . . . . . . . . . . 15 𝒫 𝐴 = 𝐴
1110eqcomi 2630 . . . . . . . . . . . . . 14 𝐴 = 𝒫 𝐴
122, 11hmeof1o 21490 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝑓:𝑋1-1-onto𝐴)
13 f1of 6099 . . . . . . . . . . . . 13 (𝑓:𝑋1-1-onto𝐴𝑓:𝑋𝐴)
14 frn 6015 . . . . . . . . . . . . 13 (𝑓:𝑋𝐴 → ran 𝑓𝐴)
1512, 13, 143syl 18 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → ran 𝑓𝐴)
1615adantr 481 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → ran 𝑓𝐴)
179, 16syl5ss 3598 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ⊆ 𝐴)
18 vex 3192 . . . . . . . . . . . 12 𝑓 ∈ V
1918imaex 7058 . . . . . . . . . . 11 (𝑓𝑥) ∈ V
2019elpw 4141 . . . . . . . . . 10 ((𝑓𝑥) ∈ 𝒫 𝐴 ↔ (𝑓𝑥) ⊆ 𝐴)
2117, 20sylibr 224 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝒫 𝐴)
222hmeoopn 21492 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → (𝑥𝐽 ↔ (𝑓𝑥) ∈ 𝒫 𝐴))
2321, 22mpbird 247 . . . . . . . 8 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ 𝑥𝑋) → 𝑥𝐽)
2423ex 450 . . . . . . 7 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥𝑋𝑥𝐽))
258, 24syl5 34 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → (𝑥 ∈ 𝒫 𝑋𝑥𝐽))
2625ssrdv 3593 . . . . 5 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
2726exlimiv 1855 . . . 4 (∃𝑓 𝑓 ∈ (𝐽Homeo𝒫 𝐴) → 𝒫 𝑋𝐽)
287, 27sylbi 207 . . 3 ((𝐽Homeo𝒫 𝐴) ≠ ∅ → 𝒫 𝑋𝐽)
296, 28sylbi 207 . 2 (𝐽 ≃ 𝒫 𝐴 → 𝒫 𝑋𝐽)
305, 29eqssd 3604 1 (𝐽 ≃ 𝒫 𝐴𝐽 = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wss 3559  c0 3896  𝒫 cpw 4135   cuni 4407   class class class wbr 4618  ran crn 5080  cima 5082  wf 5848  1-1-ontowf1o 5851  (class class class)co 6610  Homeochmeo 21479  chmph 21480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-1o 7512  df-map 7811  df-top 20631  df-topon 20648  df-cn 20954  df-hmeo 21481  df-hmph 21482
This theorem is referenced by: (None)
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