MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmeores Structured version   Visualization version   GIF version

Theorem hmeores 21776
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 21765 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 472 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 hmeores.1 . . . . 5 𝑋 = 𝐽
43cnrest 21291 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
52, 4sylancom 704 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
6 cntop2 21247 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
72, 6syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ Top)
8 eqid 2760 . . . . . 6 𝐾 = 𝐾
98toptopon 20924 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
107, 9sylib 208 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
11 df-ima 5279 . . . . . 6 (𝐹𝑌) = ran (𝐹𝑌)
1211eqimss2i 3801 . . . . 5 ran (𝐹𝑌) ⊆ (𝐹𝑌)
1312a1i 11 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ (𝐹𝑌))
14 imassrn 5635 . . . . 5 (𝐹𝑌) ⊆ ran 𝐹
153, 8cnf 21252 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
162, 15syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹:𝑋 𝐾)
17 frn 6214 . . . . . 6 (𝐹:𝑋 𝐾 → ran 𝐹 𝐾)
1816, 17syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran 𝐹 𝐾)
1914, 18syl5ss 3755 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ⊆ 𝐾)
20 cnrest2 21292 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝐹𝑌) ⊆ (𝐹𝑌) ∧ (𝐹𝑌) ⊆ 𝐾) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
2110, 13, 19, 20syl3anc 1477 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
225, 21mpbid 222 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))))
23 hmeocnvcn 21766 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2423adantr 472 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
258, 3cnf 21252 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐽) → 𝐹: 𝐾𝑋)
26 ffun 6209 . . . . 5 (𝐹: 𝐾𝑋 → Fun 𝐹)
27 funcnvres 6128 . . . . 5 (Fun 𝐹(𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
2824, 25, 26, 274syl 19 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
298cnrest 21291 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹𝑌) ⊆ 𝐾) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
3024, 19, 29syl2anc 696 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
3128, 30eqeltrd 2839 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
32 cntop1 21246 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
332, 32syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ Top)
343toptopon 20924 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3533, 34sylib 208 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
36 dfdm4 5471 . . . . . 6 dom (𝐹𝑌) = ran (𝐹𝑌)
37 fssres 6231 . . . . . . . 8 ((𝐹:𝑋 𝐾𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3816, 37sylancom 704 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
39 fdm 6212 . . . . . . 7 ((𝐹𝑌):𝑌 𝐾 → dom (𝐹𝑌) = 𝑌)
4038, 39syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → dom (𝐹𝑌) = 𝑌)
4136, 40syl5eqr 2808 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) = 𝑌)
42 eqimss 3798 . . . . 5 (ran (𝐹𝑌) = 𝑌 → ran (𝐹𝑌) ⊆ 𝑌)
4341, 42syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ 𝑌)
44 simpr 479 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝑌𝑋)
45 cnrest2 21292 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran (𝐹𝑌) ⊆ 𝑌𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4635, 43, 44, 45syl3anc 1477 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4731, 46mpbid 222 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌)))
48 ishmeo 21764 . 2 ((𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))) ↔ ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))) ∧ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4922, 47, 48sylanbrc 701 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wss 3715   cuni 4588  ccnv 5265  dom cdm 5266  ran crn 5267  cres 5268  cima 5269  Fun wfun 6043  wf 6045  cfv 6049  (class class class)co 6813  t crest 16283  Topctop 20900  TopOnctopon 20917   Cn ccn 21230  Homeochmeo 21758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-fin 8125  df-fi 8482  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cn 21233  df-hmeo 21760
This theorem is referenced by:  cvmsss2  31563
  Copyright terms: Public domain W3C validator