Theorem hmeores 23831

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 23820	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		hmeores.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	3	cnrest 23345	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))
5	2, 4	sylancom 597	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))
6		cntop2 23301	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
7	2, 6	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
8		eqid 2762	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	toptopon 22977	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	7, 9	sylib 220	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		df-ima 5660	. . . . . 6 ⊢ (𝐹 “ 𝑌) = ran (𝐹 ↾ 𝑌)
12	11	eqimss2i 3997	. . . . 5 ⊢ ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌)
13	12	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌))
14		imassrn 6060	. . . . 5 ⊢ (𝐹 “ 𝑌) ⊆ ran 𝐹
15	3, 8	cnf 23306	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
16	2, 15	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
17	16	frnd 6700	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran 𝐹 ⊆ ∪ 𝐾)
18	14, 17	sstrid 3947	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 “ 𝑌) ⊆ ∪ 𝐾)
19		cnrest2 23346	. . . 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 1390	. . 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 234	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn (𝐾 ↾t (𝐹 “ 𝑌))))
22		hmeocnvcn 23821	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
23	22	adantr 484	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
24	8, 3	cnf 23306	. . . . 5 ⊢ (◡𝐹 ∈ (𝐾 Cn 𝐽) → ◡𝐹:∪ 𝐾⟶𝑋)
25		ffun 6694	. . . . 5 ⊢ (◡𝐹:∪ 𝐾⟶𝑋 → Fun ◡𝐹)
26		funcnvres 6599	. . . . 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 23345	. . . . 5 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹 “ 𝑌) ⊆ ∪ 𝐾) → (◡𝐹 ↾ (𝐹 “ 𝑌)) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
29	23, 18, 28	syl2anc 593	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (◡𝐹 ↾ (𝐹 “ 𝑌)) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
30	27, 29	eqeltrd 2862	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
31		cntop1 23300	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
32	2, 31	syl 17	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ Top)
33	3	toptopon 22977	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
34	32, 33	sylib 220	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
35		dfdm4 5871	. . . . . 6 ⊢ dom (𝐹 ↾ 𝑌) = ran ◡(𝐹 ↾ 𝑌)
36		fssres 6730	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶∪ 𝐾 ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌):𝑌⟶∪ 𝐾)
37	16, 36	sylancom 597	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌):𝑌⟶∪ 𝐾)
38	37	fdmd 6702	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → dom (𝐹 ↾ 𝑌) = 𝑌)
39	35, 38	eqtr3id 2811	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran ◡(𝐹 ↾ 𝑌) = 𝑌)
40		eqimss 3994	. . . . 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 488	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝑌 ⊆ 𝑋)
43		cnrest2 23346	. . . 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 1390	. . 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 234	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn (𝐽 ↾t 𝑌)))
46		ishmeo 23819	. 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 592	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ⊆ wss 3904  ∪ cuni 4865  ^◡ccnv 5646  dom cdm 5647  ran crn 5648   ↾ cres 5649   “ cima 5650  Fun wfun 6515  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ↾_t crest 17449  Topctop 22953  TopOnctopon 22970   Cn ccn 23284  Homeochmeo 23813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-map 8810  df-en 8928  df-fin 8931  df-fi 9357  df-rest 17451  df-topgen 17472  df-top 22954  df-topon 22971  df-bases 23006  df-cn 23287  df-hmeo 23815

This theorem is referenced by:  cvmsss2  35624