Theorem hmeoopn 22917

Description: Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
hmeoopn.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
hmeoopn	⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐹 “ 𝐴) ∈ 𝐾))

Proof of Theorem hmeoopn

Step	Hyp	Ref	Expression
1		hmeoima 22916	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ∈ 𝐽) → (𝐹 “ 𝐴) ∈ 𝐾)
2	1	ex 413	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → (𝐴 ∈ 𝐽 → (𝐹 “ 𝐴) ∈ 𝐾))
3	2	adantr 481	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 → (𝐹 “ 𝐴) ∈ 𝐾))
4		hmeocn 22911	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
5		cnima 22416	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ∈ 𝐾) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽)
6	5	ex 413	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐹 “ 𝐴) ∈ 𝐾 → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽))
7	4, 6	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ((𝐹 “ 𝐴) ∈ 𝐾 → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽))
8	7	adantr 481	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ 𝐾 → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽))
9		hmeoopn.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
10		eqid 2738	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
11	9, 10	hmeof1o 22915	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
12		f1of1 6715	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–1-1→∪ 𝐾)
13	11, 12	syl 17	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1→∪ 𝐾)
14		f1imacnv 6732	. . . . 5 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
15	13, 14	sylan 580	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
16	15	eleq1d 2823	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((◡𝐹 “ (𝐹 “ 𝐴)) ∈ 𝐽 ↔ 𝐴 ∈ 𝐽))
17	8, 16	sylibd 238	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ 𝐾 → 𝐴 ∈ 𝐽))
18	3, 17	impbid 211	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐹 “ 𝐴) ∈ 𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ∪ cuni 4839  ^◡ccnv 5588   “ cima 5592  –1-1→wf1 6430  –1-1-onto→wf1o 6432  (class class class)co 7275   Cn ccn 22375  Homeochmeo 22904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-top 22043  df-topon 22060  df-cn 22378  df-hmeo 22906

This theorem is referenced by:  hmphdis  22947