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Theorem hmeontr 23793
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 23784 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 480 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 imassrn 6091 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
4 hmeoopn.1 . . . . . . . . 9 𝑋 = 𝐽
5 eqid 2735 . . . . . . . . 9 𝐾 = 𝐾
64, 5hmeof1o 23788 . . . . . . . 8 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
76adantr 480 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋1-1-onto 𝐾)
8 f1ofo 6856 . . . . . . 7 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋onto 𝐾)
9 forn 6824 . . . . . . 7 (𝐹:𝑋onto 𝐾 → ran 𝐹 = 𝐾)
107, 8, 93syl 18 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ran 𝐹 = 𝐾)
113, 10sseqtrid 4048 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹𝐴) ⊆ 𝐾)
125cnntri 23295 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝐴) ⊆ 𝐾) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))))
132, 11, 12syl2anc 584 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))))
14 f1of1 6848 . . . . . . 7 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
157, 14syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋1-1 𝐾)
16 f1imacnv 6865 . . . . . 6 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1715, 16sylancom 588 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1817fveq2d 6911 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))) = ((int‘𝐽)‘𝐴))
1913, 18sseqtrd 4036 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20 f1ofun 6851 . . . . 5 (𝐹:𝑋1-1-onto 𝐾 → Fun 𝐹)
217, 20syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → Fun 𝐹)
22 cntop2 23265 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
232, 22syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐾 ∈ Top)
245ntrss3 23084 . . . . . 6 ((𝐾 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐾) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ 𝐾)
2523, 11, 24syl2anc 584 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ 𝐾)
2625, 10sseqtrrd 4037 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ ran 𝐹)
27 funimass1 6650 . . . 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 584 . . 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 23785 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
314cnntri 23295 . . . 4 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
3230, 31sylan 580 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
33 imacnvcnv 6228 . . 3 (𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34 imacnvcnv 6228 . . . 4 (𝐹𝐴) = (𝐹𝐴)
3534fveq2i 6910 . . 3 ((int‘𝐾)‘(𝐹𝐴)) = ((int‘𝐾)‘(𝐹𝐴))
3632, 33, 353sstr3g 4040 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
3729, 36eqssd 4013 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wss 3963   cuni 4912  ccnv 5688  ran crn 5690  cima 5692  Fun wfun 6557  1-1wf1 6560  ontowfo 6561  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  Topctop 22915  intcnt 23041   Cn ccn 23248  Homeochmeo 23777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867  df-top 22916  df-topon 22933  df-ntr 23044  df-cn 23251  df-hmeo 23779
This theorem is referenced by: (None)
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