Theorem opncldf1 22235

Description: A bijection useful for converting statements about open sets to statements about closed sets and vice versa. (Contributed by Jeff Hankins, 27-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)

Hypotheses

Ref	Expression
opncldf.1	⊢ 𝑋 = ∪ 𝐽
opncldf.2	⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))

Assertion

Ref	Expression
opncldf1	⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑢,𝐽   𝑢,𝑋,𝑥

Allowed substitution hint:   𝐹(𝑢)

Proof of Theorem opncldf1

Step	Hyp	Ref	Expression
1		opncldf.2	. 2 ⊢ 𝐹 = (𝑢 ∈ 𝐽 ↦ (𝑋 ∖ 𝑢))
2		opncldf.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
3	2	opncld 22184	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → (𝑋 ∖ 𝑢) ∈ (Clsd‘𝐽))
4	2	cldopn 22182	. . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽)
5	4	adantl 482	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽)
6	2	cldss 22180	. . . . . . 7 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋)
7	6	ad2antll 726	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 ⊆ 𝑋)
8		dfss4 4192	. . . . . 6 ⊢ (𝑥 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥)
9	7, 8	sylib 217	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥)
10	9	eqcomd 2744	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋 ∖ 𝑥)))
11		difeq2 4051	. . . . 5 ⊢ (𝑢 = (𝑋 ∖ 𝑥) → (𝑋 ∖ 𝑢) = (𝑋 ∖ (𝑋 ∖ 𝑥)))
12	11	eqeq2d 2749	. . . 4 ⊢ (𝑢 = (𝑋 ∖ 𝑥) → (𝑥 = (𝑋 ∖ 𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋 ∖ 𝑥))))
13	10, 12	syl5ibrcom 246	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) → 𝑥 = (𝑋 ∖ 𝑢)))
14	2	eltopss 22056	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
15	14	adantrr 714	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 ⊆ 𝑋)
16		dfss4 4192	. . . . . 6 ⊢ (𝑢 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢)
17	15, 16	sylib 217	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢)
18	17	eqcomd 2744	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋 ∖ 𝑢)))
19		difeq2 4051	. . . . 5 ⊢ (𝑥 = (𝑋 ∖ 𝑢) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ 𝑢)))
20	19	eqeq2d 2749	. . . 4 ⊢ (𝑥 = (𝑋 ∖ 𝑢) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑢 = (𝑋 ∖ (𝑋 ∖ 𝑢))))
21	18, 20	syl5ibrcom 246	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑥 = (𝑋 ∖ 𝑢) → 𝑢 = (𝑋 ∖ 𝑥)))
22	13, 21	impbid 211	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑥 = (𝑋 ∖ 𝑢)))
23	1, 3, 5, 22	f1ocnv2d 7522	1 ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ⊆ wss 3887  ∪ cuni 4839   ↦ cmpt 5157  ^◡ccnv 5588  –1-1-onto→wf1o 6432  ‘cfv 6433  Topctop 22042  Clsdccld 22167

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-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-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-cld 22170

This theorem is referenced by:  opncldf3  22237  cmpfi  22559