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Theorem hmeontr 21382
 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 21373 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 480 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 imassrn 5396 . . . . . 6 (𝐹𝐴) ⊆ ran 𝐹
4 hmeoopn.1 . . . . . . . . 9 𝑋 = 𝐽
5 eqid 2610 . . . . . . . . 9 𝐾 = 𝐾
64, 5hmeof1o 21377 . . . . . . . 8 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋1-1-onto 𝐾)
76adantr 480 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋1-1-onto 𝐾)
8 f1ofo 6057 . . . . . . 7 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋onto 𝐾)
9 forn 6031 . . . . . . 7 (𝐹:𝑋onto 𝐾 → ran 𝐹 = 𝐾)
107, 8, 93syl 18 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ran 𝐹 = 𝐾)
113, 10syl5sseq 3616 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹𝐴) ⊆ 𝐾)
125cnntri 20885 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹𝐴) ⊆ 𝐾) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))))
132, 11, 12syl2anc 691 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))))
14 f1of1 6049 . . . . . . 7 (𝐹:𝑋1-1-onto 𝐾𝐹:𝑋1-1 𝐾)
157, 14syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐹:𝑋1-1 𝐾)
16 f1imacnv 6066 . . . . . 6 ((𝐹:𝑋1-1 𝐾𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1715, 16sylancom 698 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ (𝐹𝐴)) = 𝐴)
1817fveq2d 6107 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐽)‘(𝐹 “ (𝐹𝐴))) = ((int‘𝐽)‘𝐴))
1913, 18sseqtrd 3604 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐾)‘(𝐹𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20 f1ofun 6052 . . . . 5 (𝐹:𝑋1-1-onto 𝐾 → Fun 𝐹)
217, 20syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → Fun 𝐹)
22 cntop2 20855 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
232, 22syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → 𝐾 ∈ Top)
245ntrss3 20674 . . . . . 6 ((𝐾 ∈ Top ∧ (𝐹𝐴) ⊆ 𝐾) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ 𝐾)
2523, 11, 24syl2anc 691 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ 𝐾)
2625, 10sseqtr4d 3605 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) ⊆ ran 𝐹)
27 funimass1 5885 . . . 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 691 . . 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 21374 . . . 4 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
314cnntri 20885 . . . 4 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
3230, 31sylan 487 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
33 imacnvcnv 5517 . . 3 (𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34 imacnvcnv 5517 . . . 4 (𝐹𝐴) = (𝐹𝐴)
3534fveq2i 6106 . . 3 ((int‘𝐾)‘(𝐹𝐴)) = ((int‘𝐾)‘(𝐹𝐴))
3632, 33, 353sstr3g 3608 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹𝐴)))
3729, 36eqssd 3585 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴𝑋) → ((int‘𝐾)‘(𝐹𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372  ◡ccnv 5037  ran crn 5039   “ cima 5041  Fun wfun 5798  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Topctop 20517  intcnt 20631   Cn ccn 20838  Homeochmeo 21366 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-top 20521  df-topon 20523  df-ntr 20634  df-cn 20841  df-hmeo 21368 This theorem is referenced by: (None)
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