ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hmeores GIF version

Theorem hmeores 12498
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 12488 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 274 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 hmeores.1 . . . . 5 𝑋 = 𝐽
43cnrest 12418 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
52, 4sylancom 416 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
6 cntop2 12385 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
72, 6syl 14 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ Top)
8 eqid 2139 . . . . . 6 𝐾 = 𝐾
98toptopon 12199 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
107, 9sylib 121 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
11 df-ima 4552 . . . . . 6 (𝐹𝑌) = ran (𝐹𝑌)
1211eqimss2i 3154 . . . . 5 ran (𝐹𝑌) ⊆ (𝐹𝑌)
1312a1i 9 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ (𝐹𝑌))
14 imassrn 4892 . . . . 5 (𝐹𝑌) ⊆ ran 𝐹
153, 8cnf 12387 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
162, 15syl 14 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹:𝑋 𝐾)
1716frnd 5282 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran 𝐹 𝐾)
1814, 17sstrid 3108 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ⊆ 𝐾)
19 cnrest2 12419 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝐹𝑌) ⊆ (𝐹𝑌) ∧ (𝐹𝑌) ⊆ 𝐾) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
2010, 13, 18, 19syl3anc 1216 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
215, 20mpbid 146 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))))
22 hmeocnvcn 12489 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2322adantr 274 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
248, 3cnf 12387 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐽) → 𝐹: 𝐾𝑋)
25 ffun 5275 . . . . 5 (𝐹: 𝐾𝑋 → Fun 𝐹)
26 funcnvres 5196 . . . . 5 (Fun 𝐹(𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
2723, 24, 25, 264syl 18 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
288cnrest 12418 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹𝑌) ⊆ 𝐾) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
2923, 18, 28syl2anc 408 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
3027, 29eqeltrd 2216 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
31 cntop1 12384 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
322, 31syl 14 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ Top)
333toptopon 12199 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3432, 33sylib 121 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
35 dfdm4 4731 . . . . . 6 dom (𝐹𝑌) = ran (𝐹𝑌)
36 fssres 5298 . . . . . . . 8 ((𝐹:𝑋 𝐾𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3716, 36sylancom 416 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3837fdmd 5279 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → dom (𝐹𝑌) = 𝑌)
3935, 38syl5eqr 2186 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) = 𝑌)
40 eqimss 3151 . . . . 5 (ran (𝐹𝑌) = 𝑌 → ran (𝐹𝑌) ⊆ 𝑌)
4139, 40syl 14 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ 𝑌)
42 simpr 109 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝑌𝑋)
43 cnrest2 12419 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran (𝐹𝑌) ⊆ 𝑌𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4434, 41, 42, 43syl3anc 1216 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4530, 44mpbid 146 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌)))
46 ishmeo 12487 . 2 ((𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))) ↔ ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))) ∧ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4721, 45, 46sylanbrc 413 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wcel 1480  wss 3071   cuni 3736  ccnv 4538  dom cdm 4539  ran crn 4540  cres 4541  cima 4542  Fun wfun 5117  wf 5119  cfv 5123  (class class class)co 5774  t crest 12134  Topctop 12178  TopOnctopon 12191   Cn ccn 12368  Homeochmeo 12483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-rest 12136  df-topgen 12155  df-top 12179  df-topon 12192  df-bases 12224  df-cn 12371  df-hmeo 12484
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator