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Theorem hmeontr 23593
Description: Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
hmeoopn.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
hmeontr ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) = (𝐹 β€œ ((intβ€˜π½)β€˜π΄)))
Proof of Theorem hmeontr
StepHypRef Expression
1 hmeocn 23584 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 480 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
3 imassrn 6070 . . . . . 6 (𝐹 β€œ 𝐴) βŠ† ran 𝐹
4 hmeoopn.1 . . . . . . . . 9 𝑋 = βˆͺ 𝐽
5 eqid 2731 . . . . . . . . 9 βˆͺ 𝐾 = βˆͺ 𝐾
64, 5hmeof1o 23588 . . . . . . . 8 (𝐹 ∈ (𝐽Homeo𝐾) β†’ 𝐹:𝑋–1-1-ontoβ†’βˆͺ 𝐾)
76adantr 480 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐹:𝑋–1-1-ontoβ†’βˆͺ 𝐾)
8 f1ofo 6840 . . . . . . 7 (𝐹:𝑋–1-1-ontoβ†’βˆͺ 𝐾 β†’ 𝐹:𝑋–ontoβ†’βˆͺ 𝐾)
9 forn 6808 . . . . . . 7 (𝐹:𝑋–ontoβ†’βˆͺ 𝐾 β†’ ran 𝐹 = βˆͺ 𝐾)
107, 8, 93syl 18 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ran 𝐹 = βˆͺ 𝐾)
113, 10sseqtrid 4034 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐹 β€œ 𝐴) βŠ† βˆͺ 𝐾)
125cnntri 23095 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 β€œ 𝐴) βŠ† βˆͺ 𝐾) β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴))) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ (𝐹 β€œ 𝐴))))
132, 11, 12syl2anc 583 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴))) βŠ† ((intβ€˜π½)β€˜(◑𝐹 β€œ (𝐹 β€œ 𝐴))))
14 f1of1 6832 . . . . . . 7 (𝐹:𝑋–1-1-ontoβ†’βˆͺ 𝐾 β†’ 𝐹:𝑋–1-1β†’βˆͺ 𝐾)
157, 14syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐹:𝑋–1-1β†’βˆͺ 𝐾)
16 f1imacnv 6849 . . . . . 6 ((𝐹:𝑋–1-1β†’βˆͺ 𝐾 ∧ 𝐴 βŠ† 𝑋) β†’ (◑𝐹 β€œ (𝐹 β€œ 𝐴)) = 𝐴)
1715, 16sylancom 587 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ (◑𝐹 β€œ (𝐹 β€œ 𝐴)) = 𝐴)
1817fveq2d 6895 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜π½)β€˜(◑𝐹 β€œ (𝐹 β€œ 𝐴))) = ((intβ€˜π½)β€˜π΄))
1913, 18sseqtrd 4022 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ (◑𝐹 β€œ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴))) βŠ† ((intβ€˜π½)β€˜π΄))
20 f1ofun 6835 . . . . 5 (𝐹:𝑋–1-1-ontoβ†’βˆͺ 𝐾 β†’ Fun 𝐹)
217, 20syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ Fun 𝐹)
22 cntop2 23065 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
232, 22syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐾 ∈ Top)
245ntrss3 22884 . . . . . 6 ((𝐾 ∈ Top ∧ (𝐹 β€œ 𝐴) βŠ† βˆͺ 𝐾) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† βˆͺ 𝐾)
2523, 11, 24syl2anc 583 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† βˆͺ 𝐾)
2625, 10sseqtrrd 4023 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† ran 𝐹)
27 funimass1 6630 . . . 4 ((Fun 𝐹 ∧ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† ran 𝐹) β†’ ((◑𝐹 β€œ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴))) βŠ† ((intβ€˜π½)β€˜π΄) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† (𝐹 β€œ ((intβ€˜π½)β€˜π΄))))
2821, 26, 27syl2anc 583 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((◑𝐹 β€œ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴))) βŠ† ((intβ€˜π½)β€˜π΄) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† (𝐹 β€œ ((intβ€˜π½)β€˜π΄))))
2919, 28mpd 15 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† (𝐹 β€œ ((intβ€˜π½)β€˜π΄)))
30 hmeocnvcn 23585 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) β†’ ◑𝐹 ∈ (𝐾 Cn 𝐽))
314cnntri 23095 . . . 4 ((◑𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 βŠ† 𝑋) β†’ (◑◑𝐹 β€œ ((intβ€˜π½)β€˜π΄)) βŠ† ((intβ€˜πΎ)β€˜(◑◑𝐹 β€œ 𝐴)))
3230, 31sylan 579 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ (◑◑𝐹 β€œ ((intβ€˜π½)β€˜π΄)) βŠ† ((intβ€˜πΎ)β€˜(◑◑𝐹 β€œ 𝐴)))
33 imacnvcnv 6205 . . 3 (◑◑𝐹 β€œ ((intβ€˜π½)β€˜π΄)) = (𝐹 β€œ ((intβ€˜π½)β€˜π΄))
34 imacnvcnv 6205 . . . 4 (◑◑𝐹 β€œ 𝐴) = (𝐹 β€œ 𝐴)
3534fveq2i 6894 . . 3 ((intβ€˜πΎ)β€˜(◑◑𝐹 β€œ 𝐴)) = ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴))
3632, 33, 353sstr3g 4026 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐹 β€œ ((intβ€˜π½)β€˜π΄)) βŠ† ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)))
3729, 36eqssd 3999 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) = (𝐹 β€œ ((intβ€˜π½)β€˜π΄)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   βŠ† wss 3948  βˆͺ cuni 4908  β—‘ccnv 5675  ran crn 5677   β€œ cima 5679  Fun wfun 6537  β€“1-1β†’wf1 6540  β€“ontoβ†’wfo 6541  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7412  Topctop 22715  intcnt 22841   Cn ccn 23048  Homeochmeo 23577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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-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 7415  df-oprab 7416  df-mpo 7417  df-map 8828  df-top 22716  df-topon 22733  df-ntr 22844  df-cn 23051  df-hmeo 23579
This theorem is referenced by: (None)
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