Theorem kur14 35432

Description: Kuratowski's closure-complement theorem. There are at most 14 sets which can be obtained by the application of the closure and complement operations to a set in a topological space. (Contributed by Mario Carneiro, 11-Feb-2015.)

Hypotheses

Ref	Expression
kur14.x	⊢ 𝑋 = ∪ 𝐽
kur14.k	⊢ 𝐾 = (cls‘𝐽)
kur14.s	⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}

Assertion

Ref	Expression
kur14	⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋

Allowed substitution hints:   𝑆(𝑥,𝑦)   𝐾(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem kur14

Step	Hyp	Ref	Expression
1		kur14.s	. . . . . 6 ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}
2		eleq1 2825	. . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → (𝐴 ∈ 𝑥 ↔ if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥))
3	2	anbi1d 632	. . . . . . . 8 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → ((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) ↔ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)))
4	3	rabbidv 3408	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)})
5	4	inteqd 4909	. . . . . 6 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)})
6	1, 5	eqtrid 2784	. . . . 5 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)})
7	6	eleq1d 2822	. . . 4 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → (𝑆 ∈ Fin ↔ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ∈ Fin))
8	6	fveq2d 6846	. . . . 5 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → (♯‘𝑆) = (♯‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}))
9	8	breq1d 5110	. . . 4 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → ((♯‘𝑆) ≤ ;14 ↔ (♯‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) ≤ ;14))
10	7, 9	anbi12d 633	. . 3 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → ((𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14) ↔ (∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (♯‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) ≤ ;14)))
11		kur14.x	. . . . . . . . . 10 ⊢ 𝑋 = ∪ 𝐽
12		unieq 4876	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ∪ 𝐽 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
13	11, 12	eqtrid 2784	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
14	13	pweqd 4573	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
15	14	pweqd 4573	. . . . . . 7 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
16	13	sseq2d 3968	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
17		sn0top 22955	. . . . . . . . . . . . . 14 ⊢ {∅} ∈ Top
18	17	elimel 4551	. . . . . . . . . . . . 13 ⊢ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19		uniexg 7695	. . . . . . . . . . . . 13 ⊢ (if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top → ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V)
20	18, 19	ax-mp 5	. . . . . . . . . . . 12 ⊢ ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ V
21	20	elpw2 5281	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
22	16, 21	bitr4di 289	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
23	22	ifbid 4505	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴 ⊆ 𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
24	23	eleq1d 2822	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
25	13	difeq1d 4079	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋 ∖ 𝑦) = (∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26		kur14.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (cls‘𝐽)
27		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
28	26, 27	eqtrid 2784	. . . . . . . . . . . 12 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
29	28	fveq1d 6844	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾‘𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
30	25, 29	preq12d 4700	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} = {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
31	30	sseq1d 3967	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ({(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
32	31	ralbidv 3161	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥))
33	24, 32	anbi12d 633	. . . . . . 7 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ((if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) ↔ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)))
34	15, 33	rabeqbidv 3419	. . . . . 6 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} = {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)})
35	34	inteqd 4909	. . . . 5 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} = ∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)})
36	35	eleq1d 2822	. . . 4 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ∈ Fin ↔ ∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin))
37	35	fveq2d 6846	. . . . 5 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (♯‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) = (♯‘∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}))
38	37	breq1d 5110	. . . 4 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ((♯‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) ≤ ;14 ↔ (♯‘∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ ;14))
39	36, 38	anbi12d 633	. . 3 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ((∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (♯‘∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}) ≤ ;14) ↔ (∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (♯‘∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ ;14)))
40		eqid 2737	. . . 4 ⊢ ∪ if(𝐽 ∈ Top, 𝐽, {∅}) = ∪ if(𝐽 ∈ Top, 𝐽, {∅})
41		eqid 2737	. . . 4 ⊢ (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42		eqid 2737	. . . 4 ⊢ ∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} = ∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}
43		0elpw 5303	. . . . . 6 ⊢ ∅ ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
44	43	elimel 4551	. . . . 5 ⊢ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
45		elpwi 4563	. . . . 5 ⊢ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
46	44, 45	ax-mp 5	. . . 4 ⊢ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅})
47	18, 40, 41, 42, 46	kur14lem10 35431	. . 3 ⊢ (∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)} ∈ Fin ∧ (♯‘∩ {𝑥 ∈ 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∣ (if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)} ⊆ 𝑥)}) ≤ ;14)
48	10, 39, 47	dedth2h 4541	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐽 ∈ Top) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14))
49	48	ancoms 458	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3401  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  ifcif 4481  𝒫 cpw 4556  {csn 4582  {cpr 4584  ∪ cuni 4865  ∩ cint 4904   class class class wbr 5100  ‘cfv 6500  Fincfn 8895  1c1 11039   ≤ cle 11179  4c4 12214  ;cdc 12619  ♯chash 14265  Topctop 22849  clsccl 22974

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-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-xnn0 12487  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-hash 14266  df-top 22850  df-topon 22867  df-cld 22975  df-ntr 22976  df-cls 22977

This theorem is referenced by: (None)