Theorem hmeores 15306

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) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌))))

Proof of Theorem hmeores

Step	Hyp	Ref	Expression
1		hmeocn 15296	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		hmeores.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	3	cnrest 15226	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))
5	2, 4	sylancom 420	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))
6		cntop2 15193	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
7	2, 6	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
8		eqid 2234	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	toptopon 15009	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	7, 9	sylib 122	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		df-ima 4767	. . . . . 6 ⊢ (𝐹 “ 𝑌) = ran (𝐹 ↾ 𝑌)
12	11	eqimss2i 3299	. . . . 5 ⊢ ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌)
13	12	a1i 9	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌))
14		imassrn 5117	. . . . 5 ⊢ (𝐹 “ 𝑌) ⊆ ran 𝐹
15	3, 8	cnf 15195	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
16	2, 15	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
17	16	frnd 5523	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran 𝐹 ⊆ ∪ 𝐾)
18	14, 17	sstrid 3253	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 “ 𝑌) ⊆ ∪ 𝐾)
19		cnrest2 15227	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾) ∧ ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌) ∧ (𝐹 “ 𝑌) ⊆ ∪ 𝐾) → ((𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾) ↔ (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn (𝐾 ↾t (𝐹 “ 𝑌)))))
20	10, 13, 18, 19	syl3anc 1274	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ((𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾) ↔ (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn (𝐾 ↾t (𝐹 “ 𝑌)))))
21	5, 20	mpbid 147	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn (𝐾 ↾t (𝐹 “ 𝑌))))
22		hmeocnvcn 15297	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
23	22	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
24	8, 3	cnf 15195	. . . . 5 ⊢ (◡𝐹 ∈ (𝐾 Cn 𝐽) → ◡𝐹:∪ 𝐾⟶𝑋)
25		ffun 5516	. . . . 5 ⊢ (◡𝐹:∪ 𝐾⟶𝑋 → Fun ◡𝐹)
26		funcnvres 5434	. . . . 5 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝑌) = (◡𝐹 ↾ (𝐹 “ 𝑌)))
27	23, 24, 25, 26	4syl 18	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) = (◡𝐹 ↾ (𝐹 “ 𝑌)))
28	8	cnrest 15226	. . . . 5 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹 “ 𝑌) ⊆ ∪ 𝐾) → (◡𝐹 ↾ (𝐹 “ 𝑌)) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
29	23, 18, 28	syl2anc 411	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (◡𝐹 ↾ (𝐹 “ 𝑌)) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
30	27, 29	eqeltrd 2311	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
31		cntop1 15192	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
32	2, 31	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ Top)
33	3	toptopon 15009	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
34	32, 33	sylib 122	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
35		dfdm4 4953	. . . . . 6 ⊢ dom (𝐹 ↾ 𝑌) = ran ◡(𝐹 ↾ 𝑌)
36		fssres 5545	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶∪ 𝐾 ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌):𝑌⟶∪ 𝐾)
37	16, 36	sylancom 420	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌):𝑌⟶∪ 𝐾)
38	37	fdmd 5520	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → dom (𝐹 ↾ 𝑌) = 𝑌)
39	35, 38	eqtr3id 2281	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran ◡(𝐹 ↾ 𝑌) = 𝑌)
40		eqimss 3296	. . . . 5 ⊢ (ran ◡(𝐹 ↾ 𝑌) = 𝑌 → ran ◡(𝐹 ↾ 𝑌) ⊆ 𝑌)
41	39, 40	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran ◡(𝐹 ↾ 𝑌) ⊆ 𝑌)
42		simpr 110	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝑌 ⊆ 𝑋)
43		cnrest2 15227	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ran ◡(𝐹 ↾ 𝑌) ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋) → (◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽) ↔ ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn (𝐽 ↾t 𝑌))))
44	34, 41, 42, 43	syl3anc 1274	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽) ↔ ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn (𝐽 ↾t 𝑌))))
45	30, 44	mpbid 147	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn (𝐽 ↾t 𝑌)))
46		ishmeo 15295	. 2 ⊢ ((𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌))) ↔ ((𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn (𝐾 ↾t (𝐹 “ 𝑌))) ∧ ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn (𝐽 ↾t 𝑌))))
47	21, 45, 46	sylanbrc 417	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌))))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   ⊆ wss 3214  ∪ cuni 3919  ^◡ccnv 4753  dom cdm 4754  ran crn 4755   ↾ cres 4756   “ cima 4757  Fun wfun 5351  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058   ↾_t crest 13536  Topctop 14988  TopOnctopon 15001   Cn ccn 15176  Homeochmeo 15291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-rest 13538  df-topgen 13557  df-top 14989  df-topon 15002  df-bases 15034  df-cn 15179  df-hmeo 15292

This theorem is referenced by: (None)