Theorem hmeontr 14492

Description: Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
hmeoopn.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
hmeontr	⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

Proof of Theorem hmeontr

Step	Hyp	Ref	Expression
1		hmeocn 14484	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		imassrn 5017	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
4		hmeoopn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2193	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	hmeof1o 14488	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
7	6	adantr 276	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
8		f1ofo 5508	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–onto→∪ 𝐾)
9		forn 5480	. . . . . . 7 ⊢ (𝐹:𝑋–onto→∪ 𝐾 → ran 𝐹 = ∪ 𝐾)
10	7, 8, 9	3syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ran 𝐹 = ∪ 𝐾)
11	3, 10	sseqtrid 3230	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ 𝐴) ⊆ ∪ 𝐾)
12	5	cnntri 14403	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))))
13	2, 11, 12	syl2anc 411	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))))
14		f1of1 5500	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–1-1→∪ 𝐾)
15	7, 14	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹:𝑋–1-1→∪ 𝐾)
16		f1imacnv 5518	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
17	15, 16	sylancom 420	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
18	17	fveq2d 5559	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))) = ((int‘𝐽)‘𝐴))
19	13, 18	sseqtrd 3218	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20		f1ofun 5503	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → Fun 𝐹)
21	7, 20	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → Fun 𝐹)
22		cntop2 14381	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
23	2, 22	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ Top)
24	5	ntrss3 14302	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ∪ 𝐾)
25	23, 11, 24	syl2anc 411	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ∪ 𝐾)
26	25, 10	sseqtrrd 3219	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ran 𝐹)
27		funimass1 5332	. . . 4 ⊢ ((Fun 𝐹 ∧ ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ran 𝐹) → ((◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘𝐴) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ (𝐹 “ ((int‘𝐽)‘𝐴))))
28	21, 26, 27	syl2anc 411	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘𝐴) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ (𝐹 “ ((int‘𝐽)‘𝐴))))
29	19, 28	mpd 13	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ (𝐹 “ ((int‘𝐽)‘𝐴)))
30		hmeocnvcn 14485	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
31	4	cnntri 14403	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ⊆ 𝑋) → (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)))
32	30, 31	sylan 283	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)))
33		imacnvcnv 5131	. . 3 ⊢ (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34		imacnvcnv 5131	. . . 4 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
35	34	fveq2i 5558	. . 3 ⊢ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)) = ((int‘𝐾)‘(𝐹 “ 𝐴))
36	32, 33, 35	3sstr3g 3222	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹 “ 𝐴)))
37	29, 36	eqssd 3197	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164   ⊆ wss 3154  ∪ cuni 3836  ^◡ccnv 4659  ran crn 4661   “ cima 4663  Fun wfun 5249  –1-1→wf1 5252  –onto→wfo 5253  –1-1-onto→wf1o 5254  ‘cfv 5255  (class class class)co 5919  Topctop 14176  intcnt 14272   Cn ccn 14364  Homeochmeo 14479

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-map 6706  df-top 14177  df-topon 14190  df-ntr 14275  df-cn 14367  df-hmeo 14480

This theorem is referenced by: (None)