Theorem hmeontr 23979

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 23970	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		imassrn 6075	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
4		hmeoopn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2765	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	hmeof1o 23974	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
7	6	adantr 486	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
8		f1ofo 6832	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–onto→∪ 𝐾)
9		forn 6799	. . . . . . 7 ⊢ (𝐹:𝑋–onto→∪ 𝐾 → ran 𝐹 = ∪ 𝐾)
10	7, 8, 9	3syl 19	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ran 𝐹 = ∪ 𝐾)
11	3, 10	sseqtrid 3980	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ 𝐴) ⊆ ∪ 𝐾)
12	5	cnntri 23480	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))))
13	2, 11, 12	syl2anc 596	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))))
14		f1of1 6823	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–1-1→∪ 𝐾)
15	7, 14	syl 18	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹:𝑋–1-1→∪ 𝐾)
16		f1imacnv 6841	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
17	15, 16	sylancom 600	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
18	17	fveq2d 6889	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))) = ((int‘𝐽)‘𝐴))
19	13, 18	sseqtrd 3974	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20		f1ofun 6826	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → Fun 𝐹)
21	7, 20	syl 18	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → Fun 𝐹)
22		cntop2 23450	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
23	2, 22	syl 18	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ Top)
24	5	ntrss3 23269	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ∪ 𝐾)
25	23, 11, 24	syl2anc 596	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ∪ 𝐾)
26	25, 10	sseqtrrd 3975	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ran 𝐹)
27		funimass1 6622	. . . 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 596	. . 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 16	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ (𝐹 “ ((int‘𝐽)‘𝐴)))
30		hmeocnvcn 23971	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
31	4	cnntri 23480	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ⊆ 𝑋) → (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)))
32	30, 31	sylan 592	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)))
33		imacnvcnv 6209	. . 3 ⊢ (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34		imacnvcnv 6209	. . . 4 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
35	34	fveq2i 6888	. . 3 ⊢ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)) = ((int‘𝐾)‘(𝐹 “ 𝐴))
36	32, 33, 35	3sstr3g 3990	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹 “ 𝐴)))
37	29, 36	eqssd 3955	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906  ∪ cuni 4874  ^◡ccnv 5662  ran crn 5664   “ cima 5666  Fun wfun 6534  –1-1→wf1 6537  –onto→wfo 6538  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7419  Topctop 23102  intcnt 23226   Cn ccn 23433  Homeochmeo 23963

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8832  df-top 23103  df-topon 23120  df-ntr 23229  df-cn 23436  df-hmeo 23965

This theorem is used by: (None)