Theorem kur14 35981

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 2849	. . . . . . . . 9 ⊢ (𝐴 = if(𝐴 ⊆ 𝑋, 𝐴, ∅) → (𝐴 ∈ 𝑥 ↔ if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥))
3	2	anbi1d 643	. . . . . . . 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 3420	. . . . . . 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 4912	. . . . . 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 2808	. . . . 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 2846	. . . 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 6889	. . . . 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 5113	. . . 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 644	. . 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 4878	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ∪ 𝐽 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
13	11, 12	eqtrid 2808	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
14	13	pweqd 4574	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
15	14	pweqd 4574	. . . . . . 7 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
16	13	sseq2d 3963	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
17		sn0top 23317	. . . . . . . . . . . . . 14 ⊢ {∅} ∈ Top
18	17	elimel 4552	. . . . . . . . . . . . 13 ⊢ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19		uniexg 7757	. . . . . . . . . . . . 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 5296	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}) ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
22	16, 21	bitr4di 292	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
23	22	ifbid 4506	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴 ⊆ 𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
24	23	eleq1d 2846	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
25	13	difeq1d 4073	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋 ∖ 𝑦) = (∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26		kur14.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (cls‘𝐽)
27		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
28	26, 27	eqtrid 2808	. . . . . . . . . . . 12 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
29	28	fveq1d 6887	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾‘𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
30	25, 29	preq12d 4702	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} = {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
31	30	sseq1d 3962	. . . . . . . . 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 3186	. . . . . . . 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 644	. . . . . . 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 3430	. . . . . 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 4912	. . . . 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 2846	. . . 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 6889	. . . . 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 5113	. . . 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 644	. . 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 2761	. . . 4 ⊢ ∪ if(𝐽 ∈ Top, 𝐽, {∅}) = ∪ if(𝐽 ∈ Top, 𝐽, {∅})
41		eqid 2761	. . . 4 ⊢ (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42		eqid 2761	. . . 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 5317	. . . . . 6 ⊢ ∅ ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
44	43	elimel 4552	. . . . 5 ⊢ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
45		elpwi 4564	. . . . 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 35980	. . 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 4542	. 2 ⊢ ((𝐴 ⊆ 𝑋 ∧ 𝐽 ∈ Top) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14))
49	48	ancoms 464	1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝑋) → (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  ifcif 4482  𝒫 cpw 4557  {csn 4584  {cpr 4586  ∪ cuni 4867  ∩ cint 4907   class class class wbr 5103  ‘cfv 6538  Fincfn 8973  1c1 11201   ≤ cle 11344  4c4 12399  ;cdc 12814  ♯chash 14474  Topctop 23211  clsccl 23336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-oadd 8480  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-dju 9982  df-card 10020  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-xnn0 12680  df-z 12694  df-dec 12815  df-uz 12966  df-fz 13640  df-hash 14475  df-top 23212  df-topon 23229  df-cld 23337  df-ntr 23338  df-cls 23339

This theorem is used by: (None)