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Theorem hmeontr 23254
Description: Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
hmeoopn.1 𝑋 = 𝐽
Assertion
Ref Expression
hmeontr ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

Proof of Theorem hmeontr
StepHypRef Expression
1 hmeocn 23245 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 482 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 imassrn 6067 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
4 hmeoopn.1 . . . . . . . . 9 𝑋 = 𝐽
5 eqid 2733 . . . . . . . . 9 𝐾 = 𝐾
64, 5hmeof1o 23249 . . . . . . . 8 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
76adantr 482 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋1-1-onto 𝐾)
8 f1ofo 6836 . . . . . . 7 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋onto 𝐾)
9 forn 6804 . . . . . . 7 (𝐹:𝑋onto 𝐾 → ran 𝐹 = 𝐾)
107, 8, 93syl 18 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ran 𝐹 = 𝐾)
113, 10sseqtrid 4032 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹𝐴) ⊆ 𝐾)
125cnntri 22756 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝐴) ⊆ 𝐾) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))))
132, 11, 12syl2anc 585 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))))
14 f1of1 6828 . . . . . . 7 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
157, 14syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋1-1 𝐾)
16 f1imacnv 6845 . . . . . 6 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1715, 16sylancom 589 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1817fveq2d 6891 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))) = ((int‘𝐽)‘𝐴))
1913, 18sseqtrd 4020 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20 f1ofun 6831 . . . . 5 (𝐹:𝑋1-1-onto 𝐾 → Fun 𝐹)
217, 20syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → Fun 𝐹)
22 cntop2 22726 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
232, 22syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐾 ∈ Top)
245ntrss3 22545 . . . . . 6 ((𝐾 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐾) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ 𝐾)
2523, 11, 24syl2anc 585 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ 𝐾)
2625, 10sseqtrrd 4021 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ ran 𝐹)
27 funimass1 6626 . . . 4 ((Fun 𝐹 ∧ ((int‘𝐾)‘(𝐹𝐴)) ⊆ ran 𝐹) → ((𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘𝐴) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((int‘𝐽)‘𝐴))))
2821, 26, 27syl2anc 585 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘𝐴) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((int‘𝐽)‘𝐴))))
2919, 28mpd 15 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ (𝐹 “ ((int‘𝐽)‘𝐴)))
30 hmeocnvcn 23246 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
314cnntri 22756 . . . 4 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
3230, 31sylan 581 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
33 imacnvcnv 6201 . . 3 (𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34 imacnvcnv 6201 . . . 4 (𝐹𝐴) = (𝐹𝐴)
3534fveq2i 6890 . . 3 ((int‘𝐾)‘(𝐹𝐴)) = ((int‘𝐾)‘(𝐹𝐴))
3632, 33, 353sstr3g 4024 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
3729, 36eqssd 3997 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3946   cuni 4906  ccnv 5673  ran crn 5675  cima 5677  Fun wfun 6533  1-1wf1 6536  ontowfo 6537  1-1-ontowf1o 6538  cfv 6539  (class class class)co 7403  Topctop 22376  intcnt 22502   Cn ccn 22709  Homeochmeo 23238
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-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
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-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7406  df-oprab 7407  df-mpo 7408  df-map 8817  df-top 22377  df-topon 22394  df-ntr 22505  df-cn 22712  df-hmeo 23240
This theorem is referenced by: (None)
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