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Theorem hmeontr 23264
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 23255 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 481 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐹 ∈ (𝐽 Cn 𝐾))
3 imassrn 6068 . . . . . 6 (𝐹 β€œ 𝐴) βŠ† ran 𝐹
4 hmeoopn.1 . . . . . . . . 9 𝑋 = βˆͺ 𝐽
5 eqid 2732 . . . . . . . . 9 βˆͺ 𝐾 = βˆͺ 𝐾
64, 5hmeof1o 23259 . . . . . . . 8 (𝐹 ∈ (𝐽Homeo𝐾) β†’ 𝐹:𝑋–1-1-ontoβ†’βˆͺ 𝐾)
76adantr 481 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐹:𝑋–1-1-ontoβ†’βˆͺ 𝐾)
8 f1ofo 6837 . . . . . . 7 (𝐹:𝑋–1-1-ontoβ†’βˆͺ 𝐾 β†’ 𝐹:𝑋–ontoβ†’βˆͺ 𝐾)
9 forn 6805 . . . . . . 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 4033 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐹 β€œ 𝐴) βŠ† βˆͺ 𝐾)
125cnntri 22766 . . . . 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 584 . . . 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 6829 . . . . . . 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 6846 . . . . . 6 ((𝐹:𝑋–1-1β†’βˆͺ 𝐾 ∧ 𝐴 βŠ† 𝑋) β†’ (◑𝐹 β€œ (𝐹 β€œ 𝐴)) = 𝐴)
1715, 16sylancom 588 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ (◑𝐹 β€œ (𝐹 β€œ 𝐴)) = 𝐴)
1817fveq2d 6892 . . . 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 4021 . . 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 6832 . . . . 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 22736 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐾 ∈ Top)
232, 22syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐾 ∈ Top)
245ntrss3 22555 . . . . . 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 584 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† βˆͺ 𝐾)
2625, 10sseqtrrd 4022 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 βŠ† 𝑋) β†’ ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴)) βŠ† ran 𝐹)
27 funimass1 6627 . . . 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 584 . . 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 23256 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) β†’ ◑𝐹 ∈ (𝐾 Cn 𝐽))
314cnntri 22766 . . . 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 580 . . 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 6202 . . 3 (◑◑𝐹 β€œ ((intβ€˜π½)β€˜π΄)) = (𝐹 β€œ ((intβ€˜π½)β€˜π΄))
34 imacnvcnv 6202 . . . 4 (◑◑𝐹 β€œ 𝐴) = (𝐹 β€œ 𝐴)
3534fveq2i 6891 . . 3 ((intβ€˜πΎ)β€˜(◑◑𝐹 β€œ 𝐴)) = ((intβ€˜πΎ)β€˜(𝐹 β€œ 𝐴))
3632, 33, 353sstr3g 4025 . 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 3998 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 396   = wceq 1541   ∈ wcel 2106   βŠ† wss 3947  βˆͺ cuni 4907  β—‘ccnv 5674  ran crn 5676   β€œ cima 5678  Fun wfun 6534  β€“1-1β†’wf1 6537  β€“ontoβ†’wfo 6538  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405  Topctop 22386  intcnt 22512   Cn ccn 22719  Homeochmeo 23248
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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
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-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-top 22387  df-topon 22404  df-ntr 22515  df-cn 22722  df-hmeo 23250
This theorem is referenced by: (None)
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