Theorem hmeontr 23684

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 23675	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2	1	adantr 480	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
3		imassrn 6019	. . . . . 6 ⊢ (𝐹 “ 𝐴) ⊆ ran 𝐹
4		hmeoopn.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝐽
5		eqid 2731	. . . . . . . . 9 ⊢ ∪ 𝐾 = ∪ 𝐾
6	4, 5	hmeof1o 23679	. . . . . . . 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 6770	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–onto→∪ 𝐾)
9		forn 6738	. . . . . . 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 3972	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ 𝐴) ⊆ ∪ 𝐾)
12	5	cnntri 23186	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))))
13	2, 11, 12	syl2anc 584	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))))
14		f1of1 6762	. . . . . . 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 6779	. . . . . 6 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
17	15, 16	sylancom 588	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
18	17	fveq2d 6826	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐽)‘(◡𝐹 “ (𝐹 “ 𝐴))) = ((int‘𝐽)‘𝐴))
19	13, 18	sseqtrd 3966	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ ((int‘𝐾)‘(𝐹 “ 𝐴))) ⊆ ((int‘𝐽)‘𝐴))
20		f1ofun 6765	. . . . 5 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → Fun 𝐹)
21	7, 20	syl 17	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → Fun 𝐹)
22		cntop2 23156	. . . . . . 7 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
23	2, 22	syl 17	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐾 ∈ Top)
24	5	ntrss3 22975	. . . . . 6 ⊢ ((𝐾 ∈ Top ∧ (𝐹 “ 𝐴) ⊆ ∪ 𝐾) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ∪ 𝐾)
25	23, 11, 24	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ∪ 𝐾)
26	25, 10	sseqtrrd 3967	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) ⊆ ran 𝐹)
27		funimass1 6563	. . . 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 584	. . 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 23676	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
31	4	cnntri 23186	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ⊆ 𝑋) → (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)))
32	30, 31	sylan 580	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)))
33		imacnvcnv 6153	. . 3 ⊢ (◡◡𝐹 “ ((int‘𝐽)‘𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴))
34		imacnvcnv 6153	. . . 4 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
35	34	fveq2i 6825	. . 3 ⊢ ((int‘𝐾)‘(◡◡𝐹 “ 𝐴)) = ((int‘𝐾)‘(𝐹 “ 𝐴))
36	32, 33, 35	3sstr3g 3982	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐹 “ ((int‘𝐽)‘𝐴)) ⊆ ((int‘𝐾)‘(𝐹 “ 𝐴)))
37	29, 36	eqssd 3947	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((int‘𝐾)‘(𝐹 “ 𝐴)) = (𝐹 “ ((int‘𝐽)‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  ∪ cuni 4856  ^◡ccnv 5613  ran crn 5615   “ cima 5617  Fun wfun 6475  –1-1→wf1 6478  –onto→wfo 6479  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  Topctop 22808  intcnt 22932   Cn ccn 23139  Homeochmeo 23668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-top 22809  df-topon 22826  df-ntr 22935  df-cn 23142  df-hmeo 23670

This theorem is referenced by: (None)