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Theorem hmphdis 23307
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 4949 . . . 4 𝐽 βŠ† 𝒫 βˆͺ 𝐽
2 hmphdis.1 . . . . 5 𝑋 = βˆͺ 𝐽
32pweqi 4618 . . . 4 𝒫 𝑋 = 𝒫 βˆͺ 𝐽
41, 3sseqtrri 4019 . . 3 𝐽 βŠ† 𝒫 𝑋
54a1i 11 . 2 (𝐽 ≃ 𝒫 𝐴 β†’ 𝐽 βŠ† 𝒫 𝑋)
6 hmph 23287 . . 3 (𝐽 ≃ 𝒫 𝐴 ↔ (𝐽Homeo𝒫 𝐴) β‰  βˆ…)
7 n0 4346 . . . 4 ((𝐽Homeo𝒫 𝐴) β‰  βˆ… ↔ βˆƒπ‘“ 𝑓 ∈ (𝐽Homeo𝒫 𝐴))
8 elpwi 4609 . . . . . . 7 (π‘₯ ∈ 𝒫 𝑋 β†’ π‘₯ βŠ† 𝑋)
9 imassrn 6070 . . . . . . . . . . 11 (𝑓 β€œ π‘₯) βŠ† ran 𝑓
10 unipw 5450 . . . . . . . . . . . . . . 15 βˆͺ 𝒫 𝐴 = 𝐴
1110eqcomi 2741 . . . . . . . . . . . . . 14 𝐴 = βˆͺ 𝒫 𝐴
122, 11hmeof1o 23275 . . . . . . . . . . . . 13 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ 𝑓:𝑋–1-1-onto→𝐴)
13 f1of 6833 . . . . . . . . . . . . 13 (𝑓:𝑋–1-1-onto→𝐴 β†’ 𝑓:π‘‹βŸΆπ΄)
14 frn 6724 . . . . . . . . . . . . 13 (𝑓:π‘‹βŸΆπ΄ β†’ ran 𝑓 βŠ† 𝐴)
1512, 13, 143syl 18 . . . . . . . . . . . 12 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ ran 𝑓 βŠ† 𝐴)
1615adantr 481 . . . . . . . . . . 11 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ ran 𝑓 βŠ† 𝐴)
179, 16sstrid 3993 . . . . . . . . . 10 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ (𝑓 β€œ π‘₯) βŠ† 𝐴)
18 vex 3478 . . . . . . . . . . . 12 𝑓 ∈ V
1918imaex 7909 . . . . . . . . . . 11 (𝑓 β€œ π‘₯) ∈ V
2019elpw 4606 . . . . . . . . . 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 23277 . . . . . . . . 9 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ (π‘₯ ∈ 𝐽 ↔ (𝑓 β€œ π‘₯) ∈ 𝒫 𝐴))
2321, 22mpbird 256 . . . . . . . 8 ((𝑓 ∈ (𝐽Homeo𝒫 𝐴) ∧ π‘₯ βŠ† 𝑋) β†’ π‘₯ ∈ 𝐽)
2423ex 413 . . . . . . 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 3988 . . . . 5 (𝑓 ∈ (𝐽Homeo𝒫 𝐴) β†’ 𝒫 𝑋 βŠ† 𝐽)
2726exlimiv 1933 . . . 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 3999 1 (𝐽 ≃ 𝒫 𝐴 β†’ 𝐽 = 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908   class class class wbr 5148  ran crn 5677   β€œ cima 5679  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  (class class class)co 7411  Homeochmeo 23264   ≃ chmph 23265
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-1o 8468  df-map 8824  df-top 22403  df-topon 22420  df-cn 22738  df-hmeo 23266  df-hmph 23267
This theorem is referenced by: (None)
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