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Theorem hmeontr 22307
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 22298 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 481 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 imassrn 5934 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
4 hmeoopn.1 . . . . . . . . 9 𝑋 = 𝐽
5 eqid 2821 . . . . . . . . 9 𝐾 = 𝐾
64, 5hmeof1o 22302 . . . . . . . 8 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
76adantr 481 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋1-1-onto 𝐾)
8 f1ofo 6616 . . . . . . 7 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋onto 𝐾)
9 forn 6587 . . . . . . 7 (𝐹:𝑋onto 𝐾 → ran 𝐹 = 𝐾)
107, 8, 93syl 18 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ran 𝐹 = 𝐾)
113, 10sseqtrid 4018 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹𝐴) ⊆ 𝐾)
125cnntri 21809 . . . . 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 6608 . . . . . . 7 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
157, 14syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋1-1 𝐾)
16 f1imacnv 6625 . . . . . 6 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1715, 16sylancom 588 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1817fveq2d 6668 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))) = ((int‘𝐽)‘𝐴))
1913, 18sseqtrd 4006 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20 f1ofun 6611 . . . . 5 (𝐹:𝑋1-1-onto 𝐾 → Fun 𝐹)
217, 20syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → Fun 𝐹)
22 cntop2 21779 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
232, 22syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐾 ∈ Top)
245ntrss3 21598 . . . . . 6 ((𝐾 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐾) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ 𝐾)
2523, 11, 24syl2anc 584 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ 𝐾)
2625, 10sseqtrrd 4007 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ ran 𝐹)
27 funimass1 6430 . . . 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 22299 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
314cnntri 21809 . . . 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 6057 . . 3 (𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34 imacnvcnv 6057 . . . 4 (𝐹𝐴) = (𝐹𝐴)
3534fveq2i 6667 . . 3 ((int‘𝐾)‘(𝐹𝐴)) = ((int‘𝐾)‘(𝐹𝐴))
3632, 33, 353sstr3g 4010 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
3729, 36eqssd 3983 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wss 3935   cuni 4832  ccnv 5548  ran crn 5550  cima 5552  Fun wfun 6343  1-1wf1 6346  ontowfo 6347  1-1-ontowf1o 6348  cfv 6349  (class class class)co 7145  Topctop 21431  intcnt 21555   Cn ccn 21762  Homeochmeo 22291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8398  df-top 21432  df-topon 21449  df-ntr 21558  df-cn 21765  df-hmeo 22293
This theorem is referenced by: (None)
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