Theorem hmeores 23658

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 23647	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		hmeores.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	3	cnrest 23172	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))
5	2, 4	sylancom 588	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))
6		cntop2 23128	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
7	2, 6	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
8		eqid 2729	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	toptopon 22804	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	7, 9	sylib 218	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		df-ima 5651	. . . . . 6 ⊢ (𝐹 “ 𝑌) = ran (𝐹 ↾ 𝑌)
12	11	eqimss2i 4008	. . . . 5 ⊢ ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌)
13	12	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌))
14		imassrn 6042	. . . . 5 ⊢ (𝐹 “ 𝑌) ⊆ ran 𝐹
15	3, 8	cnf 23133	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
16	2, 15	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
17	16	frnd 6696	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran 𝐹 ⊆ ∪ 𝐾)
18	14, 17	sstrid 3958	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 “ 𝑌) ⊆ ∪ 𝐾)
19		cnrest2 23173	. . . 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 1373	. . 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 232	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn (𝐾 ↾t (𝐹 “ 𝑌))))
22		hmeocnvcn 23648	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
23	22	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
24	8, 3	cnf 23133	. . . . 5 ⊢ (◡𝐹 ∈ (𝐾 Cn 𝐽) → ◡𝐹:∪ 𝐾⟶𝑋)
25		ffun 6691	. . . . 5 ⊢ (◡𝐹:∪ 𝐾⟶𝑋 → Fun ◡𝐹)
26		funcnvres 6594	. . . . 5 ⊢ (Fun ◡𝐹 → ◡(𝐹 ↾ 𝑌) = (◡𝐹 ↾ (𝐹 “ 𝑌)))
27	23, 24, 25, 26	4syl 19	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) = (◡𝐹 ↾ (𝐹 “ 𝑌)))
28	8	cnrest 23172	. . . . 5 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹 “ 𝑌) ⊆ ∪ 𝐾) → (◡𝐹 ↾ (𝐹 “ 𝑌)) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
29	23, 18, 28	syl2anc 584	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (◡𝐹 ↾ (𝐹 “ 𝑌)) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
30	27, 29	eqeltrd 2828	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
31		cntop1 23127	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
32	2, 31	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ Top)
33	3	toptopon 22804	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
34	32, 33	sylib 218	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
35		dfdm4 5859	. . . . . 6 ⊢ dom (𝐹 ↾ 𝑌) = ran ◡(𝐹 ↾ 𝑌)
36		fssres 6726	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶∪ 𝐾 ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌):𝑌⟶∪ 𝐾)
37	16, 36	sylancom 588	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌):𝑌⟶∪ 𝐾)
38	37	fdmd 6698	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → dom (𝐹 ↾ 𝑌) = 𝑌)
39	35, 38	eqtr3id 2778	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran ◡(𝐹 ↾ 𝑌) = 𝑌)
40		eqimss 4005	. . . . 5 ⊢ (ran ◡(𝐹 ↾ 𝑌) = 𝑌 → ran ◡(𝐹 ↾ 𝑌) ⊆ 𝑌)
41	39, 40	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran ◡(𝐹 ↾ 𝑌) ⊆ 𝑌)
42		simpr 484	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝑌 ⊆ 𝑋)
43		cnrest2 23173	. . . 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 1373	. . 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 232	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn (𝐽 ↾t 𝑌)))
46		ishmeo 23646	. 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 583	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3914  ∪ cuni 4871  ^◡ccnv 5637  dom cdm 5638  ran crn 5639   ↾ cres 5640   “ cima 5641  Fun wfun 6505  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↾_t crest 17383  Topctop 22780  TopOnctopon 22797   Cn ccn 23111  Homeochmeo 23640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-map 8801  df-en 8919  df-fin 8922  df-fi 9362  df-rest 17385  df-topgen 17406  df-top 22781  df-topon 22798  df-bases 22833  df-cn 23114  df-hmeo 23642

This theorem is referenced by:  cvmsss2  35261