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Theorem hmeores 22920
Description: The restriction of a homeomorphism is a homeomorphism. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
hmeores.1 𝑋 = 𝐽
Assertion
Ref Expression
hmeores ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))))

Proof of Theorem hmeores
StepHypRef Expression
1 hmeocn 22909 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 481 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 hmeores.1 . . . . 5 𝑋 = 𝐽
43cnrest 22434 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
52, 4sylancom 588 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
6 cntop2 22390 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
72, 6syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ Top)
8 eqid 2738 . . . . . 6 𝐾 = 𝐾
98toptopon 22064 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
107, 9sylib 217 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
11 df-ima 5604 . . . . . 6 (𝐹𝑌) = ran (𝐹𝑌)
1211eqimss2i 3981 . . . . 5 ran (𝐹𝑌) ⊆ (𝐹𝑌)
1312a1i 11 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ (𝐹𝑌))
14 imassrn 5982 . . . . 5 (𝐹𝑌) ⊆ ran 𝐹
153, 8cnf 22395 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
162, 15syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹:𝑋 𝐾)
1716frnd 6610 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran 𝐹 𝐾)
1814, 17sstrid 3933 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ⊆ 𝐾)
19 cnrest2 22435 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝐹𝑌) ⊆ (𝐹𝑌) ∧ (𝐹𝑌) ⊆ 𝐾) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
2010, 13, 18, 19syl3anc 1370 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
215, 20mpbid 231 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))))
22 hmeocnvcn 22910 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2322adantr 481 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
248, 3cnf 22395 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐽) → 𝐹: 𝐾𝑋)
25 ffun 6605 . . . . 5 (𝐹: 𝐾𝑋 → Fun 𝐹)
26 funcnvres 6514 . . . . 5 (Fun 𝐹(𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
2723, 24, 25, 264syl 19 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
288cnrest 22434 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹𝑌) ⊆ 𝐾) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
2923, 18, 28syl2anc 584 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
3027, 29eqeltrd 2839 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
31 cntop1 22389 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
322, 31syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ Top)
333toptopon 22064 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3432, 33sylib 217 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
35 dfdm4 5806 . . . . . 6 dom (𝐹𝑌) = ran (𝐹𝑌)
36 fssres 6642 . . . . . . . 8 ((𝐹:𝑋 𝐾𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3716, 36sylancom 588 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3837fdmd 6613 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → dom (𝐹𝑌) = 𝑌)
3935, 38eqtr3id 2792 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) = 𝑌)
40 eqimss 3978 . . . . 5 (ran (𝐹𝑌) = 𝑌 → ran (𝐹𝑌) ⊆ 𝑌)
4139, 40syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ 𝑌)
42 simpr 485 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝑌𝑋)
43 cnrest2 22435 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran (𝐹𝑌) ⊆ 𝑌𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4434, 41, 42, 43syl3anc 1370 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4530, 44mpbid 231 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌)))
46 ishmeo 22908 . 2 ((𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))) ↔ ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))) ∧ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4721, 45, 46sylanbrc 583 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wss 3888   cuni 4841  ccnv 5590  dom cdm 5591  ran crn 5592  cres 5593  cima 5594  Fun wfun 6429  wf 6431  cfv 6435  (class class class)co 7277  t crest 17129  Topctop 22040  TopOnctopon 22057   Cn ccn 22373  Homeochmeo 22902
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5211  ax-sep 5225  ax-nul 5232  ax-pow 5290  ax-pr 5354  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3433  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4259  df-if 4462  df-pw 4537  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4842  df-int 4882  df-iun 4928  df-br 5077  df-opab 5139  df-mpt 5160  df-tr 5194  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-ord 6271  df-on 6272  df-lim 6273  df-suc 6274  df-iota 6393  df-fun 6437  df-fn 6438  df-f 6439  df-f1 6440  df-fo 6441  df-f1o 6442  df-fv 6443  df-ov 7280  df-oprab 7281  df-mpo 7282  df-om 7713  df-1st 7831  df-2nd 7832  df-map 8615  df-en 8732  df-fin 8735  df-fi 9168  df-rest 17131  df-topgen 17152  df-top 22041  df-topon 22058  df-bases 22094  df-cn 22376  df-hmeo 22904
This theorem is referenced by:  cvmsss2  33233
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