Theorem hmeontr 23798

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 23789	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		imassrn 6100	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
4		hmeoopn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2740	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	hmeof1o 23793	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
7	6	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
8		f1ofo 6869	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–onto→∪ 𝐾)
9		forn 6837	. . . . . . 7 ⊢ (𝐹:𝑋–onto→∪ 𝐾 → ran 𝐹 = ∪ 𝐾)
10	7, 8, 9	3syl 18	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ran 𝐹 = ∪ 𝐾)
11	3, 10	sseqtrid 4061	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ 𝐴) ⊆ ∪ 𝐾)
12	5	cnntri 23300	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))))
13	2, 11, 12	syl2anc 583	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))))
14		f1of1 6861	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–1-1→∪ 𝐾)
15	7, 14	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹:𝑋–1-1→∪ 𝐾)
16		f1imacnv 6878	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
17	15, 16	sylancom 587	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
18	17	fveq2d 6924	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))) = ((int‘𝐽)‘𝐴))
19	13, 18	sseqtrd 4049	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20		f1ofun 6864	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → Fun 𝐹)
21	7, 20	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → Fun 𝐹)
22		cntop2 23270	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
23	2, 22	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ Top)
24	5	ntrss3 23089	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ∪ 𝐾)
25	23, 11, 24	syl2anc 583	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ∪ 𝐾)
26	25, 10	sseqtrrd 4050	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ran 𝐹)
27		funimass1 6660	. . . 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 583	. . 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 15	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ (𝐹 “ ((int‘𝐽)‘𝐴)))
30		hmeocnvcn 23790	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
31	4	cnntri 23300	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ⊆ 𝑋) → (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)))
32	30, 31	sylan 579	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)))
33		imacnvcnv 6237	. . 3 ⊢ (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34		imacnvcnv 6237	. . . 4 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
35	34	fveq2i 6923	. . 3 ⊢ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)) = ((int‘𝐾)‘(𝐹 “ 𝐴))
36	32, 33, 35	3sstr3g 4053	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹 “ 𝐴)))
37	29, 36	eqssd 4026	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ∪ cuni 4931  ^◡ccnv 5699  ran crn 5701   “ cima 5703  Fun wfun 6567  –1-1→wf1 6570  –onto→wfo 6571  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Topctop 22920  intcnt 23046   Cn ccn 23253  Homeochmeo 23782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-top 22921  df-topon 22938  df-ntr 23049  df-cn 23256  df-hmeo 23784

This theorem is referenced by: (None)