Theorem kur14 32456

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 2898	. . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → (𝐴 ∈ 𝑥 ↔ if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥))
3	2	anbi1d 631	. . . . . . . 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 3479	. . . . . . 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 4872	. . . . . 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	syl5eq 2866	. . . . 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 2895	. . . 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 6667	. . . . 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 5067	. . . 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 632	. . 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 4838	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ∪ 𝐽 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
13	11, 12	syl5eq 2866	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
14	13	pweqd 4542	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
15	14	pweqd 4542	. . . . . . 7 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
16	13	sseq2d 3997	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
17		sn0top 21599	. . . . . . . . . . . . . 14 ⊢ {∅} ∈ Top
18	17	elimel 4532	. . . . . . . . . . . . 13 ⊢ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19		uniexg 7458	. . . . . . . . . . . . 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 5239	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
22	16, 21	syl6bbr 291	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
23	22	ifbid 4487	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴 ⊆ 𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
24	23	eleq1d 2895	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
25	13	difeq1d 4096	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋 ∖ 𝑦) = (∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26		kur14.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (cls‘𝐽)
27		fveq2 6663	. . . . . . . . . . . . 13 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
28	26, 27	syl5eq 2866	. . . . . . . . . . . 12 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
29	28	fveq1d 6665	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾‘𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
30	25, 29	preq12d 4669	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} = {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
31	30	sseq1d 3996	. . . . . . . . 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 3195	. . . . . . . 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 632	. . . . . . 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 3484	. . . . . 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 4872	. . . . 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 2895	. . . 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 6667	. . . . 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 5067	. . . 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 632	. . 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 2819	. . . 4 ⊢ ∪ if(𝐽 ∈ Top, 𝐽, {∅}) = ∪ if(𝐽 ∈ Top, 𝐽, {∅})
41		eqid 2819	. . . 4 ⊢ (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42		eqid 2819	. . . 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 5247	. . . . . 6 ⊢ ∅ ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
44	43	elimel 4532	. . . . 5 ⊢ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
45		elpwi 4549	. . . . 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 32455	. . 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 4522	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐽 ∈ Top) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14))
49	48	ancoms 461	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  {crab 3140  Vcvv 3493   ∖ cdif 3931   ⊆ wss 3934  ∅c0 4289  ifcif 4465  𝒫 cpw 4537  {csn 4559  {cpr 4561  ∪ cuni 4830  ∩ cint 4867   class class class wbr 5057  ‘cfv 6348  Fincfn 8501  1c1 10530   ≤ cle 10668  4c4 11686  ;cdc 12090  ♯chash 13682  Topctop 21493  clsccl 21618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-xnn0 11960  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12885  df-hash 13683  df-top 21494  df-topon 21511  df-cld 21619  df-ntr 21620  df-cls 21621

This theorem is referenced by: (None)