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Theorem hmphdis 23300
Description: Homeomorphisms preserve topological discreteness. (Contributed by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
hmphdis.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
hmphdis (𝐽 ≃ 𝒫 𝐴 β†’ 𝐽 = 𝒫 𝑋)
Proof of Theorem hmphdis
Dummy variables π‘₯ 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwuni 4950 . . . 4 𝐽 βŠ† 𝒫 βˆͺ 𝐽
2 hmphdis.1 . . . . 5 𝑋 = βˆͺ 𝐽
32pweqi 4619 . . . 4 𝒫 𝑋 = 𝒫 βˆͺ 𝐽
41, 3sseqtrri 4020 . . 3 𝐽 βŠ† 𝒫 𝑋
54a1i 11 . 2 (𝐽 ≃ 𝒫 𝐴 β†’ 𝐽 βŠ† 𝒫 𝑋)
6 hmph 23280 . . 3 (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) β‰  βˆ…)
7 n0 4347 . . . 4 ((𝐽Homeo𝒫 𝐴) β‰  βˆ… ↔ βˆƒπ‘“ 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8 elpwi 4610 . . . . . . 7 (π‘₯ ∈ 𝒫 𝑋 β†’ π‘₯ βŠ† 𝑋)
9 imassrn 6071 . . . . . . . . . . 11 (𝑓 β€œ π‘₯) βŠ† ran 𝑓
10 unipw 5451 . . . . . . . . . . . . . . 15 βˆͺ 𝒫 𝐴 = 𝐴
1110eqcomi 2742 . . . . . . . . . . . . . 14 𝐴 = βˆͺ 𝒫 𝐴
122, 11hmeof1o 23268 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ 𝑓:𝑋–1-1-onto→𝐴)
13 f1of 6834 . . . . . . . . . . . . 13 (𝑓:𝑋–1-1-onto→𝐴 β†’ 𝑓:π‘‹βŸΆπ΄)
14 frn 6725 . . . . . . . . . . . . 13 (𝑓:π‘‹βŸΆπ΄ β†’ ran 𝑓 βŠ† 𝐴)
1512, 13, 143syl 18 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ ran 𝑓 βŠ† 𝐴)
1615adantr 482 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ ran 𝑓 βŠ† 𝐴)
179, 16sstrid 3994 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ (𝑓 β€œ π‘₯) βŠ† 𝐴)
18 vex 3479 . . . . . . . . . . . 12 𝑓 ∈ V
1918imaex 7907 . . . . . . . . . . 11 (𝑓 β€œ π‘₯) ∈ V
2019elpw 4607 . . . . . . . . . 10 ((𝑓 β€œ π‘₯) ∈ 𝒫 𝐴 ↔ (𝑓 β€œ π‘₯) βŠ† 𝐴)
2117, 20sylibr 233 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ (𝑓 β€œ π‘₯) ∈ 𝒫 𝐴)
222hmeoopn 23270 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ (π‘₯ ∈ 𝐽 ↔ (𝑓 β€œ π‘₯) ∈ 𝒫 𝐴))
2321, 22mpbird 257 . . . . . . . 8 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ π‘₯ ∈ 𝐽)
2423ex 414 . . . . . . 7 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ (π‘₯ βŠ† 𝑋 β†’ π‘₯ ∈ 𝐽))
258, 24syl5 34 . . . . . 6 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ (π‘₯ ∈ 𝒫 𝑋 β†’ π‘₯ ∈ 𝐽))
2625ssrdv 3989 . . . . 5 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ 𝒫 𝑋 βŠ† 𝐽)
2726exlimiv 1934 . . . 4 (βˆƒπ‘“ 𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ 𝒫 𝑋 βŠ† 𝐽)
287, 27sylbi 216 . . 3 ((𝐽Homeo𝒫 𝐴) β‰  βˆ… β†’ 𝒫 𝑋 βŠ† 𝐽)
296, 28sylbi 216 . 2 (𝐽 ≃ 𝒫 𝐴 β†’ 𝒫 𝑋 βŠ† 𝐽)
305, 29eqssd 4000 1 (𝐽 ≃ 𝒫 𝐴 β†’ 𝐽 = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909   class class class wbr 5149  ran crn 5678   β€œ cima 5680  βŸΆwf 6540  β€“1-1-ontoβ†’wf1o 6543  (class class class)co 7409  Homeochmeo 23257   ≃ chmph 23258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-1o 8466  df-map 8822  df-top 22396  df-topon 22413  df-cn 22731  df-hmeo 23259  df-hmph 23260
This theorem is referenced by: (None)
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