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Theorem hmeores 22354
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 22343 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 484 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 hmeores.1 . . . . 5 𝑋 = 𝐽
43cnrest 21868 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
52, 4sylancom 591 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
6 cntop2 21824 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
72, 6syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ Top)
8 eqid 2821 . . . . . 6 𝐾 = 𝐾
98toptopon 21500 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
107, 9sylib 221 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
11 df-ima 5541 . . . . . 6 (𝐹𝑌) = ran (𝐹𝑌)
1211eqimss2i 4002 . . . . 5 ran (𝐹𝑌) ⊆ (𝐹𝑌)
1312a1i 11 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ (𝐹𝑌))
14 imassrn 5913 . . . . 5 (𝐹𝑌) ⊆ ran 𝐹
153, 8cnf 21829 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
162, 15syl 17 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹:𝑋 𝐾)
1716frnd 6494 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran 𝐹 𝐾)
1814, 17sstrid 3954 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ⊆ 𝐾)
19 cnrest2 21869 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝐹𝑌) ⊆ (𝐹𝑌) ∧ (𝐹𝑌) ⊆ 𝐾) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
2010, 13, 18, 19syl3anc 1368 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
215, 20mpbid 235 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))))
22 hmeocnvcn 22344 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2322adantr 484 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
248, 3cnf 21829 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐽) → 𝐹: 𝐾𝑋)
25 ffun 6490 . . . . 5 (𝐹: 𝐾𝑋 → Fun 𝐹)
26 funcnvres 6405 . . . . 5 (Fun 𝐹(𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
2723, 24, 25, 264syl 19 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
288cnrest 21868 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹𝑌) ⊆ 𝐾) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
2923, 18, 28syl2anc 587 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
3027, 29eqeltrd 2912 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
31 cntop1 21823 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
322, 31syl 17 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ Top)
333toptopon 21500 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3432, 33sylib 221 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
35 dfdm4 5737 . . . . . 6 dom (𝐹𝑌) = ran (𝐹𝑌)
36 fssres 6517 . . . . . . . 8 ((𝐹:𝑋 𝐾𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3716, 36sylancom 591 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3837fdmd 6496 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → dom (𝐹𝑌) = 𝑌)
3935, 38syl5eqr 2870 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) = 𝑌)
40 eqimss 3999 . . . . 5 (ran (𝐹𝑌) = 𝑌 → ran (𝐹𝑌) ⊆ 𝑌)
4139, 40syl 17 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ 𝑌)
42 simpr 488 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝑌𝑋)
43 cnrest2 21869 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran (𝐹𝑌) ⊆ 𝑌𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4434, 41, 42, 43syl3anc 1368 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4530, 44mpbid 235 . 2 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌)))
46 ishmeo 22342 . 2 ((𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))) ↔ ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))) ∧ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4721, 45, 46sylanbrc 586 1 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wss 3910   cuni 4811  ccnv 5527  dom cdm 5528  ran crn 5529  cres 5530  cima 5531  Fun wfun 6322  wf 6324  cfv 6328  (class class class)co 7130  t crest 16672  Topctop 21476  TopOnctopon 21493   Cn ccn 21807  Homeochmeo 22336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-oadd 8081  df-er 8264  df-map 8383  df-en 8485  df-fin 8488  df-fi 8851  df-rest 16674  df-topgen 16695  df-top 21477  df-topon 21494  df-bases 21529  df-cn 21810  df-hmeo 22338
This theorem is referenced by:  cvmsss2  32528
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