Theorem opncldf1 23250

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 23199	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → (𝑋 ∖ 𝑢) ∈ (Clsd‘𝐽))
4	2	cldopn 23197	. . 3 ⊢ (𝑥 ∈ (Clsd‘𝐽) → (𝑋 ∖ 𝑥) ∈ 𝐽)
5	4	adantl 486	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ (Clsd‘𝐽)) → (𝑋 ∖ 𝑥) ∈ 𝐽)
6	2	cldss 23195	. . . . . . 7 ⊢ (𝑥 ∈ (Clsd‘𝐽) → 𝑥 ⊆ 𝑋)
7	6	ad2antll 741	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 ⊆ 𝑋)
8		dfss4 4222	. . . . . 6 ⊢ (𝑥 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥)
9	7, 8	sylib 221	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋 ∖ 𝑥)) = 𝑥)
10	9	eqcomd 2769	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑥 = (𝑋 ∖ (𝑋 ∖ 𝑥)))
11		difeq2 4075	. . . . 5 ⊢ (𝑢 = (𝑋 ∖ 𝑥) → (𝑋 ∖ 𝑢) = (𝑋 ∖ (𝑋 ∖ 𝑥)))
12	11	eqeq2d 2774	. . . 4 ⊢ (𝑢 = (𝑋 ∖ 𝑥) → (𝑥 = (𝑋 ∖ 𝑢) ↔ 𝑥 = (𝑋 ∖ (𝑋 ∖ 𝑥))))
13	10, 12	syl5ibrcom 250	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) → 𝑥 = (𝑋 ∖ 𝑢)))
14	2	eltopss 23073	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑢 ∈ 𝐽) → 𝑢 ⊆ 𝑋)
15	14	adantrr 729	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 ⊆ 𝑋)
16		dfss4 4222	. . . . . 6 ⊢ (𝑢 ⊆ 𝑋 ↔ (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢)
17	15, 16	sylib 221	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑋 ∖ (𝑋 ∖ 𝑢)) = 𝑢)
18	17	eqcomd 2769	. . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → 𝑢 = (𝑋 ∖ (𝑋 ∖ 𝑢)))
19		difeq2 4075	. . . . 5 ⊢ (𝑥 = (𝑋 ∖ 𝑢) → (𝑋 ∖ 𝑥) = (𝑋 ∖ (𝑋 ∖ 𝑢)))
20	19	eqeq2d 2774	. . . 4 ⊢ (𝑥 = (𝑋 ∖ 𝑢) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑢 = (𝑋 ∖ (𝑋 ∖ 𝑢))))
21	18, 20	syl5ibrcom 250	. . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑥 = (𝑋 ∖ 𝑢) → 𝑢 = (𝑋 ∖ 𝑥)))
22	13, 21	impbid 215	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝑢 ∈ 𝐽 ∧ 𝑥 ∈ (Clsd‘𝐽))) → (𝑢 = (𝑋 ∖ 𝑥) ↔ 𝑥 = (𝑋 ∖ 𝑢)))
23	1, 3, 5, 22	f1ocnv2d 7663	1 ⊢ (𝐽 ∈ Top → (𝐹:𝐽–1-1-onto→(Clsd‘𝐽) ∧ ◡𝐹 = (𝑥 ∈ (Clsd‘𝐽) ↦ (𝑋 ∖ 𝑥))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ∖ cdif 3902   ⊆ wss 3905  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5660  –1-1-onto→wf1o 6535  ‘cfv 6536  Topctop 23059  Clsdccld 23182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-top 23060  df-cld 23185

This theorem is used by:  opncldf3  23252  cmpfi  23574