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Theorem hmeores 13109
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 13099 . . . . 5 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
21adantr 274 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3 hmeores.1 . . . . 5 𝑋 = 𝐽
43cnrest 13029 . . . 4 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
52, 4sylancom 418 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾))
6 cntop2 12996 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
72, 6syl 14 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ Top)
8 eqid 2170 . . . . . 6 𝐾 = 𝐾
98toptopon 12810 . . . . 5 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
107, 9sylib 121 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐾 ∈ (TopOn‘ 𝐾))
11 df-ima 4624 . . . . . 6 (𝐹𝑌) = ran (𝐹𝑌)
1211eqimss2i 3204 . . . . 5 ran (𝐹𝑌) ⊆ (𝐹𝑌)
1312a1i 9 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) ⊆ (𝐹𝑌))
14 imassrn 4964 . . . . 5 (𝐹𝑌) ⊆ ran 𝐹
153, 8cnf 12998 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋 𝐾)
162, 15syl 14 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹:𝑋 𝐾)
1716frnd 5357 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran 𝐹 𝐾)
1814, 17sstrid 3158 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ⊆ 𝐾)
19 cnrest2 13030 . . . 4 ((𝐾 ∈ (TopOn‘ 𝐾) ∧ ran (𝐹𝑌) ⊆ (𝐹𝑌) ∧ (𝐹𝑌) ⊆ 𝐾) → ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn 𝐾) ↔ (𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌)))))
2010, 13, 18, 19syl3anc 1233 . . 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 13100 . . . . . 6 (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐾 Cn 𝐽))
2322adantr 274 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐹 ∈ (𝐾 Cn 𝐽))
248, 3cnf 12998 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐽) → 𝐹: 𝐾𝑋)
25 ffun 5350 . . . . 5 (𝐹: 𝐾𝑋 → Fun 𝐹)
26 funcnvres 5271 . . . . 5 (Fun 𝐹(𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
2723, 24, 25, 264syl 18 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) = (𝐹 ↾ (𝐹𝑌)))
288cnrest 13029 . . . . 5 ((𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹𝑌) ⊆ 𝐾) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
2923, 18, 28syl2anc 409 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹 ↾ (𝐹𝑌)) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
3027, 29eqeltrd 2247 . . 3 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽))
31 cntop1 12995 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
322, 31syl 14 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ Top)
333toptopon 12810 . . . . 5 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
3432, 33sylib 121 . . . 4 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → 𝐽 ∈ (TopOn‘𝑋))
35 dfdm4 4803 . . . . . 6 dom (𝐹𝑌) = ran (𝐹𝑌)
36 fssres 5373 . . . . . . . 8 ((𝐹:𝑋 𝐾𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3716, 36sylancom 418 . . . . . . 7 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → (𝐹𝑌):𝑌 𝐾)
3837fdmd 5354 . . . . . 6 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → dom (𝐹𝑌) = 𝑌)
3935, 38eqtr3id 2217 . . . . 5 ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌𝑋) → ran (𝐹𝑌) = 𝑌)
40 eqimss 3201 . . . . 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 13030 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ ran (𝐹𝑌) ⊆ 𝑌𝑌𝑋) → ((𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn 𝐽) ↔ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4434, 41, 42, 43syl3anc 1233 . . 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 13098 . 2 ((𝐹𝑌) ∈ ((𝐽t 𝑌)Homeo(𝐾t (𝐹𝑌))) ↔ ((𝐹𝑌) ∈ ((𝐽t 𝑌) Cn (𝐾t (𝐹𝑌))) ∧ (𝐹𝑌) ∈ ((𝐾t (𝐹𝑌)) Cn (𝐽t 𝑌))))
4721, 45, 46sylanbrc 415 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 1348  wcel 2141  wss 3121   cuni 3796  ccnv 4610  dom cdm 4611  ran crn 4612  cres 4613  cima 4614  Fun wfun 5192  wf 5194  cfv 5198  (class class class)co 5853  t crest 12579  Topctop 12789  TopOnctopon 12802   Cn ccn 12979  Homeochmeo 13094
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-rest 12581  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-hmeo 13095
This theorem is referenced by: (None)
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