Theorem opncldf1 23045

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 22994	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → (𝑋 ∖ 𝑢) ∈ (Clsd‘𝐽))
4	2	cldopn 22992	. . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽)
5	4	adantl 481	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽)
6	2	cldss 22990	. . . . . . 7 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋)
7	6	ad2antll 730	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 ⊆ 𝑋)
8		dfss4 4223	. . . . . 6 ⊢ (𝑥 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥)
9	7, 8	sylib 218	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥)
10	9	eqcomd 2743	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋 ∖ 𝑥)))
11		difeq2 4074	. . . . 5 ⊢ (𝑢 = (𝑋 ∖ 𝑥) → (𝑋 ∖ 𝑢) = (𝑋 ∖ (𝑋 ∖ 𝑥)))
12	11	eqeq2d 2748	. . . 4 ⊢ (𝑢 = (𝑋 ∖ 𝑥) → (𝑥 = (𝑋 ∖ 𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋 ∖ 𝑥))))
13	10, 12	syl5ibrcom 247	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) → 𝑥 = (𝑋 ∖ 𝑢)))
14	2	eltopss 22868	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
15	14	adantrr 718	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 ⊆ 𝑋)
16		dfss4 4223	. . . . . 6 ⊢ (𝑢 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢)
17	15, 16	sylib 218	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢)
18	17	eqcomd 2743	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋 ∖ 𝑢)))
19		difeq2 4074	. . . . 5 ⊢ (𝑥 = (𝑋 ∖ 𝑢) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ 𝑢)))
20	19	eqeq2d 2748	. . . 4 ⊢ (𝑥 = (𝑋 ∖ 𝑢) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑢 = (𝑋 ∖ (𝑋 ∖ 𝑢))))
21	18, 20	syl5ibrcom 247	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑥 = (𝑋 ∖ 𝑢) → 𝑢 = (𝑋 ∖ 𝑥)))
22	13, 21	impbid 212	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑥 = (𝑋 ∖ 𝑢)))
23	1, 3, 5, 22	f1ocnv2d 7623	1 ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3900   ⊆ wss 3903  ∪ cuni 4865   ↦ cmpt 5181  ^◡ccnv 5633  –1-1-onto→wf1o 6501  ‘cfv 6502  Topctop 22854  Clsdccld 22977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-top 22855  df-cld 22980

This theorem is referenced by:  opncldf3  23047  cmpfi  23369