Theorem hmeocld 13952

Description: Homeomorphisms preserve closedness. (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
hmeocld	⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))

Proof of Theorem hmeocld

Step	Hyp	Ref	Expression
1		hmeocnvcn 13946	. . . 4 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
2	1	adantr 276	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ◡𝐹 ∈ (𝐾 Cn 𝐽))
3		imacnvcnv 5095	. . . . 5 ⊢ (◡◡𝐹 “ 𝐴) = (𝐹 “ 𝐴)
4		cnclima 13863	. . . . 5 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (◡◡𝐹 “ 𝐴) ∈ (Clsd‘𝐾))
5	3, 4	eqeltrrid 2265	. . . 4 ⊢ ((◡𝐹 ∈ (𝐾 Cn 𝐽) ∧ 𝐴 ∈ (Clsd‘𝐽)) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾))
6	5	ex 115	. . 3 ⊢ (◡𝐹 ∈ (𝐾 Cn 𝐽) → (𝐴 ∈ (Clsd‘𝐽) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))
7	2, 6	syl 14	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) → (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))
8		hmeocn 13945	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
9	8	adantr 276	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → 𝐹 ∈ (𝐽 Cn 𝐾))
10		cnclima 13863	. . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽))
11	10	ex 115	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽)))
12	9, 11	syl 14	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → (◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽)))
13		hmeoopn.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
14		eqid 2177	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
15	13, 14	hmeof1o 13949	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→∪ 𝐾)
16		f1of1 5462	. . . . . 6 ⊢ (𝐹:𝑋–1-1-onto→∪ 𝐾 → 𝐹:𝑋–1-1→∪ 𝐾)
17	15, 16	syl 14	. . . . 5 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1→∪ 𝐾)
18		f1imacnv 5480	. . . . 5 ⊢ ((𝐹:𝑋–1-1→∪ 𝐾 ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
19	17, 18	sylan 283	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝐴)) = 𝐴)
20	19	eleq1d 2246	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((◡𝐹 “ (𝐹 “ 𝐴)) ∈ (Clsd‘𝐽) ↔ 𝐴 ∈ (Clsd‘𝐽)))
21	12, 20	sylibd 149	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → ((𝐹 “ 𝐴) ∈ (Clsd‘𝐾) → 𝐴 ∈ (Clsd‘𝐽)))
22	7, 21	impbid 129	1 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝐴 ⊆ 𝑋) → (𝐴 ∈ (Clsd‘𝐽) ↔ (𝐹 “ 𝐴) ∈ (Clsd‘𝐾)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ⊆ wss 3131  ∪ cuni 3811  ^◡ccnv 4627   “ cima 4631  –1-1→wf1 5215  –1-1-onto→wf1o 5217  ‘cfv 5218  (class class class)co 5878  Clsdccld 13732   Cn ccn 13825  Homeochmeo 13940

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-map 6653  df-top 13638  df-topon 13651  df-cld 13735  df-cn 13828  df-hmeo 13941

This theorem is referenced by: (None)