Theorem hmeontr 15304

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 15296	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		imassrn 5117	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
4		hmeoopn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2234	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	hmeof1o 15300	. . . . . . . 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 5626	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–onto→∪ 𝐾)
9		forn 5598	. . . . . . 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 3292	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ 𝐴) ⊆ ∪ 𝐾)
12	5	cnntri 15215	. . . . 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 5618	. . . . . . 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 5636	. . . . . 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 5679	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))) = ((int‘𝐽)‘𝐴))
19	13, 18	sseqtrd 3280	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20		f1ofun 5621	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → Fun 𝐹)
21	7, 20	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → Fun 𝐹)
22		cntop2 15193	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
23	2, 22	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ Top)
24	5	ntrss3 15114	. . . . . 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 3281	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ran 𝐹)
27		funimass1 5438	. . . 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 15297	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
31	4	cnntri 15215	. . . 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 5232	. . 3 ⊢ (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34		imacnvcnv 5232	. . . 4 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
35	34	fveq2i 5678	. . 3 ⊢ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)) = ((int‘𝐾)‘(𝐹 “ 𝐴))
36	32, 33, 35	3sstr3g 3284	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹 “ 𝐴)))
37	29, 36	eqssd 3259	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205   ⊆ wss 3214  ∪ cuni 3919  ^◡ccnv 4753  ran crn 4755   “ cima 4757  Fun wfun 5351  –1-1→wf1 5354  –onto→wfo 5355  –1-1-onto→wf1o 5356  ‘cfv 5357  (class class class)co 6058  Topctop 14988  intcnt 15084   Cn ccn 15176  Homeochmeo 15291

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-top 14989  df-topon 15002  df-ntr 15087  df-cn 15179  df-hmeo 15292

This theorem is referenced by: (None)