Theorem kur14 35238

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 2822	. . . . . . . . 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 3423	. . . . . . 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 4927	. . . . . 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 2782	. . . . 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 2819	. . . 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 6880	. . . . 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 5129	. . . 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 4894	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → ∪ 𝐽 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
13	11, 12	eqtrid 2782	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝑋 = ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
14	13	pweqd 4592	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝑋 = 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
15	14	pweqd 4592	. . . . . . 7 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝒫 𝒫 𝑋 = 𝒫 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}))
16	13	sseq2d 3991	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐴 ⊆ 𝑋 ↔ 𝐴 ⊆ ∪ if(𝐽 ∈ Top, 𝐽, {∅})))
17		sn0top 22937	. . . . . . . . . . . . . 14 ⊢ {∅} ∈ Top
18	17	elimel 4570	. . . . . . . . . . . . 13 ⊢ if(𝐽 ∈ Top, 𝐽, {∅}) ∈ Top
19		uniexg 7734	. . . . . . . . . . . . 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 5304	. . . . . . . . . . 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 4524	. . . . . . . . 9 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → if(𝐴 ⊆ 𝑋, 𝐴, ∅) = if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅))
24	23	eleq1d 2819	. . . . . . . 8 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (if(𝐴 ⊆ 𝑋, 𝐴, ∅) ∈ 𝑥 ↔ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝑥))
25	13	difeq1d 4100	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝑋 ∖ 𝑦) = (∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦))
26		kur14.k	. . . . . . . . . . . . 13 ⊢ 𝐾 = (cls‘𝐽)
27		fveq2 6876	. . . . . . . . . . . . 13 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (cls‘𝐽) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
28	26, 27	eqtrid 2782	. . . . . . . . . . . 12 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → 𝐾 = (cls‘if(𝐽 ∈ Top, 𝐽, {∅})))
29	28	fveq1d 6878	. . . . . . . . . . 11 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → (𝐾‘𝑦) = ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦))
30	25, 29	preq12d 4717	. . . . . . . . . 10 ⊢ (𝐽 = if(𝐽 ∈ Top, 𝐽, {∅}) → {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} = {(∪ if(𝐽 ∈ Top, 𝐽, {∅}) ∖ 𝑦), ((cls‘if(𝐽 ∈ Top, 𝐽, {∅}))‘𝑦)})
31	30	sseq1d 3990	. . . . . . . . 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 3163	. . . . . . . 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 3434	. . . . . 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 4927	. . . . 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 2819	. . . 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 6880	. . . . 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 5129	. . . 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 2735	. . . 4 ⊢ ∪ if(𝐽 ∈ Top, 𝐽, {∅}) = ∪ if(𝐽 ∈ Top, 𝐽, {∅})
41		eqid 2735	. . . 4 ⊢ (cls‘if(𝐽 ∈ Top, 𝐽, {∅})) = (cls‘if(𝐽 ∈ Top, 𝐽, {∅}))
42		eqid 2735	. . . 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 5326	. . . . . 6 ⊢ ∅ ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
44	43	elimel 4570	. . . . 5 ⊢ if(𝐴 ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅}), 𝐴, ∅) ∈ 𝒫 ∪ if(𝐽 ∈ Top, 𝐽, {∅})
45		elpwi 4582	. . . . 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 35237	. . 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 4560	. 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 1540   ∈ wcel 2108  ∀wral 3051  {crab 3415  Vcvv 3459   ∖ cdif 3923   ⊆ wss 3926  ∅c0 4308  ifcif 4500  𝒫 cpw 4575  {csn 4601  {cpr 4603  ∪ cuni 4883  ∩ cint 4922   class class class wbr 5119  ‘cfv 6531  Fincfn 8959  1c1 11130   ≤ cle 11270  4c4 12297  ;cdc 12708  ♯chash 14348  Topctop 22831  clsccl 22956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9915  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-xnn0 12575  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-hash 14349  df-top 22832  df-topon 22849  df-cld 22957  df-ntr 22958  df-cls 22959

This theorem is referenced by: (None)