Theorem hmeores 13074

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 13064	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 274	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		hmeores.1	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
4	3	cnrest 12994	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))
5	2, 4	sylancom 418	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn 𝐾))
6		cntop2 12961	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
7	2, 6	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ Top)
8		eqid 2170	. . . . . 6 ⊢ ∪ 𝐾 = ∪ 𝐾
9	8	toptopon 12775	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
10	7, 9	sylib 121	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
11		df-ima 4622	. . . . . 6 ⊢ (𝐹 “ 𝑌) = ran (𝐹 ↾ 𝑌)
12	11	eqimss2i 3204	. . . . 5 ⊢ ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌)
13	12	a1i 9	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran (𝐹 ↾ 𝑌) ⊆ (𝐹 “ 𝑌))
14		imassrn 4962	. . . . 5 ⊢ (𝐹 “ 𝑌) ⊆ ran 𝐹
15	3, 8	cnf 12963	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
16	2, 15	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
17	16	frnd 5355	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran 𝐹 ⊆ ∪ 𝐾)
18	14, 17	sstrid 3158	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 “ 𝑌) ⊆ ∪ 𝐾)
19		cnrest2 12995	. . . 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 1233	. . 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 146	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌) Cn (𝐾 ↾t (𝐹 “ 𝑌))))
22		hmeocnvcn 13065	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
23	22	adantr 274	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
24	8, 3	cnf 12963	. . . . 5 ⊢ (◡𝐹 ∈ (𝐾 Cn 𝐽) → ◡𝐹:∪ 𝐾⟶𝑋)
25		ffun 5348	. . . . 5 ⊢ (◡𝐹:∪ 𝐾⟶𝑋 → Fun ◡𝐹)
26		funcnvres 5269	. . . . 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 12994	. . . . 5 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ (𝐹 “ 𝑌) ⊆ ∪ 𝐾) → (◡𝐹 ↾ (𝐹 “ 𝑌)) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
29	23, 18, 28	syl2anc 409	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (◡𝐹 ↾ (𝐹 “ 𝑌)) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
30	27, 29	eqeltrd 2247	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn 𝐽))
31		cntop1 12960	. . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
32	2, 31	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ Top)
33	3	toptopon 12775	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
34	32, 33	sylib 121	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
35		dfdm4 4801	. . . . . 6 ⊢ dom (𝐹 ↾ 𝑌) = ran ◡(𝐹 ↾ 𝑌)
36		fssres 5371	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶∪ 𝐾 ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌):𝑌⟶∪ 𝐾)
37	16, 36	sylancom 418	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌):𝑌⟶∪ 𝐾)
38	37	fdmd 5352	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → dom (𝐹 ↾ 𝑌) = 𝑌)
39	35, 38	eqtr3id 2217	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ran ◡(𝐹 ↾ 𝑌) = 𝑌)
40		eqimss 3201	. . . . 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 109	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → 𝑌 ⊆ 𝑋)
43		cnrest2 12995	. . . 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 1233	. . 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 146	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → ◡(𝐹 ↾ 𝑌) ∈ ((𝐾 ↾t (𝐹 “ 𝑌)) Cn (𝐽 ↾t 𝑌)))
46		ishmeo 13063	. 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 415	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑌 ⊆ 𝑋) → (𝐹 ↾ 𝑌) ∈ ((𝐽 ↾t 𝑌)Homeo(𝐾 ↾t (𝐹 “ 𝑌))))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348   ∈ wcel 2141   ⊆ wss 3121  ∪ cuni 3794  ^◡ccnv 4608  dom cdm 4609  ran crn 4610   ↾ cres 4611   “ cima 4612  Fun wfun 5190  ⟶wf 5192  ‘cfv 5196  (class class class)co 5851   ↾_t crest 12572  Topctop 12754  TopOnctopon 12767   Cn ccn 12944  Homeochmeo 13059

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 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519

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 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-map 6626  df-rest 12574  df-topgen 12593  df-top 12755  df-topon 12768  df-bases 12800  df-cn 12947  df-hmeo 13060

This theorem is referenced by: (None)