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Theorem cmpsublem 23389

Description: Lemma for cmpsub 23390. (Contributed by Jeff Hankins, 28-Jun-2009.)

**Hypothesis**
Ref	Expression
cmpsub.1	⊢ 𝑋 = ∪ 𝐽

**Assertion**
Ref	Expression
cmpsublem	⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∀𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)(∪ (𝐽 ↾_t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))

Distinct variable groups: 𝑐,𝑑,𝑠,𝑡,𝐽 𝑆,𝑐,𝑑,𝑠,𝑡 𝑋,𝑐,𝑑,𝑠,𝑡

Proof of Theorem cmpsublem Dummy variables 𝑥 𝑦 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rabexg 5272	. . . . . . 7 ⊢ (𝐽 ∈ Top → {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ V)
2	1	ad2antrr 732	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ V)
3		ssrab2 4018	. . . . . . 7 ⊢ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ⊆ 𝐽
4		elpwg 4539	. . . . . . 7 ⊢ ({𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ V → ({𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ 𝒫 𝐽 ↔ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ⊆ 𝐽))
5	3, 4	mpbiri 259	. . . . . 6 ⊢ ({𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ V → {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ 𝒫 𝐽)
6	2, 5	syl 17	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ 𝒫 𝐽)
7		unieq 4856	. . . . . . . 8 ⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∪ 𝑐 = ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})
8	7	sseq2d 3954	. . . . . . 7 ⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑆 ⊆ ∪ 𝑐 ↔ 𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))
9		pweq 4550	. . . . . . . . 9 ⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → 𝒫 𝑐 = 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})
10	9	ineq1d 4155	. . . . . . . 8 ⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝒫 𝑐 ∩ Fin) = (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin))
11	10	rexeqdv 3299	. . . . . . 7 ⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑 ↔ ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑))
12	8, 11	imbi12d 345	. . . . . 6 ⊢ (𝑐 = {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ((𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) ↔ (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
13	12	rspcva 3565	. . . . 5 ⊢ (({𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∈ 𝒫 𝐽 ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)) → (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑))
14	6, 13	sylan 586	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑)) → (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑))
15	14	ex 413	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑)))
16		cmpsub.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
17	16	restuni 23152	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 = ∪ (𝐽 ↾_t 𝑆))
18	17	adantr 481	. . . . . 6 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → 𝑆 = ∪ (𝐽 ↾_t 𝑆))
19	18	eqeq1d 2742	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → (𝑆 = ∪ 𝑠 ↔ ∪ (𝐽 ↾_t 𝑆) = ∪ 𝑠))
20		velpw 4541	. . . . . . . . . . 11 ⊢ (𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆) ↔ 𝑠 ⊆ (𝐽 ↾_t 𝑆))
21		eleq2 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑆 = ∪ 𝑠 → (𝑡 ∈ 𝑆 ↔ 𝑡 ∈ ∪ 𝑠))
22		eluni 4848	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ ∪ 𝑠 ↔ ∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠))
23	21, 22	bitrdi 288	. . . . . . . . . . . . . 14 ⊢ (𝑆 = ∪ 𝑠 → (𝑡 ∈ 𝑆 ↔ ∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠)))
24	23	adantl 482	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑡 ∈ 𝑆 ↔ ∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠)))
25		ssel 3916	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ⊆ (𝐽 ↾_t 𝑆) → (𝑢 ∈ 𝑠 → 𝑢 ∈ (𝐽 ↾_t 𝑆)))
26	16	sseq2i 3951	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑆 ⊆ 𝑋 ↔ 𝑆 ⊆ ∪ 𝐽)
27		uniexg 7690	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐽 ∈ Top → ∪ 𝐽 ∈ V)
28		ssexg 5258	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ⊆ ∪ 𝐽 ∧ ∪ 𝐽 ∈ V) → 𝑆 ∈ V)
29	28	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((∪ 𝐽 ∈ V ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ∈ V)
30	27, 29	sylan 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → 𝑆 ∈ V)
31	26, 30	sylan2b 600	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ V)
32		elrest 17388	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ V) → (𝑢 ∈ (𝐽 ↾_t 𝑆) ↔ ∃𝑤 ∈ 𝐽 𝑢 = (𝑤 ∩ 𝑆)))
33	31, 32	syldan 597	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑢 ∈ (𝐽 ↾_t 𝑆) ↔ ∃𝑤 ∈ 𝐽 𝑢 = (𝑤 ∩ 𝑆)))
34		inss1 4172	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑤 ∩ 𝑆) ⊆ 𝑤
35		sseq1 3947	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑢 = (𝑤 ∩ 𝑆) → (𝑢 ⊆ 𝑤 ↔ (𝑤 ∩ 𝑆) ⊆ 𝑤))
36	34, 35	mpbiri 259	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑢 = (𝑤 ∩ 𝑆) → 𝑢 ⊆ 𝑤)
37	36	sselda 3922	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑢 = (𝑤 ∩ 𝑆) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ 𝑤)
38	37	3ad2antl3 1194	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ 𝑤)
39	38	3adant2 1137	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → 𝑡 ∈ 𝑤)
40		ineq1 4149	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 = 𝑤 → (𝑦 ∩ 𝑆) = (𝑤 ∩ 𝑆))
41	40	eleq1d 2825	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑤 → ((𝑦 ∩ 𝑆) ∈ 𝑠 ↔ (𝑤 ∩ 𝑆) ∈ 𝑠))
42		simp12 1211	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → 𝑤 ∈ 𝐽)
43		eleq1 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑢 = (𝑤 ∩ 𝑆) → (𝑢 ∈ 𝑠 ↔ (𝑤 ∩ 𝑆) ∈ 𝑠))
44	43	biimpa 477	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑢 = (𝑤 ∩ 𝑆) ∧ 𝑢 ∈ 𝑠) → (𝑤 ∩ 𝑆) ∈ 𝑠)
45	44	3ad2antl3 1194	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠) → (𝑤 ∩ 𝑆) ∈ 𝑠)
46	45	3adant3 1138	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → (𝑤 ∩ 𝑆) ∈ 𝑠)
47	41, 42, 46	elrabd 3638	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → 𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})
48		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑤 ∈ V
49		eleq2 2829	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣 = 𝑤 → (𝑡 ∈ 𝑣 ↔ 𝑡 ∈ 𝑤))
50		eleq1 2828	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑣 = 𝑤 → (𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ 𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))
51	49, 50	anbi12d 638	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑣 = 𝑤 → ((𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) ↔ (𝑡 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))
52	48, 51	spcev 3551	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑡 ∈ 𝑤 ∧ 𝑤 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))
53	39, 47, 52	syl2anc 590	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) ∧ 𝑢 ∈ 𝑠 ∧ 𝑡 ∈ 𝑢) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))
54	53	3exp 1125	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑢 = (𝑤 ∩ 𝑆)) → (𝑢 ∈ 𝑠 → (𝑡 ∈ 𝑢 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))
55	54	rexlimdv3a 3145	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∃𝑤 ∈ 𝐽 𝑢 = (𝑤 ∩ 𝑆) → (𝑢 ∈ 𝑠 → (𝑡 ∈ 𝑢 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))))
56	33, 55	sylbid 241	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑢 ∈ (𝐽 ↾_t 𝑆) → (𝑢 ∈ 𝑠 → (𝑡 ∈ 𝑢 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))))
57	56	com23 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑢 ∈ 𝑠 → (𝑢 ∈ (𝐽 ↾_t 𝑆) → (𝑡 ∈ 𝑢 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))))
58	57	com4l 92	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ 𝑠 → (𝑢 ∈ (𝐽 ↾_t 𝑆) → (𝑡 ∈ 𝑢 → ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))))
59	25, 58	sylcom 30	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 ⊆ (𝐽 ↾_t 𝑆) → (𝑢 ∈ 𝑠 → (𝑡 ∈ 𝑢 → ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))))
60	59	com24 95	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ⊆ (𝐽 ↾_t 𝑆) → ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑡 ∈ 𝑢 → (𝑢 ∈ 𝑠 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))))
61	60	impcom 408	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾_t 𝑆)) → (𝑡 ∈ 𝑢 → (𝑢 ∈ 𝑠 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))
62	61	impd 411	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾_t 𝑆)) → ((𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))
63	62	exlimdv 1940	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾_t 𝑆)) → (∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))
64	63	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (∃𝑢(𝑡 ∈ 𝑢 ∧ 𝑢 ∈ 𝑠) → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))
65	24, 64	sylbid 241	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑡 ∈ 𝑆 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))
66	65	ex 413	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ⊆ (𝐽 ↾_t 𝑆)) → (𝑆 = ∪ 𝑠 → (𝑡 ∈ 𝑆 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))
67	20, 66	sylan2b 600	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → (𝑆 = ∪ 𝑠 → (𝑡 ∈ 𝑆 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))))
68	67	imp 407	. . . . . . . . 9 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑡 ∈ 𝑆 → ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})))
69		eluni 4848	. . . . . . . . 9 ⊢ (𝑡 ∈ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ ∃𝑣(𝑡 ∈ 𝑣 ∧ 𝑣 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))
70	68, 69	imbitrrdi 253	. . . . . . . 8 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑡 ∈ 𝑆 → 𝑡 ∈ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))
71	70	ssrdv 3928	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → 𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})
72		pm2.27 42	. . . . . . . . 9 ⊢ (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑))
73		elin 3906	. . . . . . . . . . 11 ⊢ (𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin) ↔ (𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin))
74		vex 3436	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑡 ∈ V
75		eqeq1 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑡 → (𝑥 = (𝑧 ∩ 𝑆) ↔ 𝑡 = (𝑧 ∩ 𝑆)))
76	75	rexbidv 3164	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑡 → (∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆) ↔ ∃𝑧 ∈ 𝑑 𝑡 = (𝑧 ∩ 𝑆)))
77	74, 76	elab 3624	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ↔ ∃𝑧 ∈ 𝑑 𝑡 = (𝑧 ∩ 𝑆))
78		velpw 4541	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ 𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠})
79		ssel 3916	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑧 ∈ 𝑑 → 𝑧 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠}))
80		ineq1 4149	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 = 𝑧 → (𝑦 ∩ 𝑆) = (𝑧 ∩ 𝑆))
81	80	eleq1d 2825	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑧 → ((𝑦 ∩ 𝑆) ∈ 𝑠 ↔ (𝑧 ∩ 𝑆) ∈ 𝑠))
82	81	elrab 3636	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ↔ (𝑧 ∈ 𝐽 ∧ (𝑧 ∩ 𝑆) ∈ 𝑠))
83		eleq1a 2835	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑧 ∩ 𝑆) ∈ 𝑠 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠))
84	82, 83	simplbiim 509	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠))
85	79, 84	syl6 35	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)))
86	85	2a1d 26	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)))))
87	86	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ⊆ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)))))
88	78, 87	sylanb 587	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)))))
89	88	3imp 1116	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → (𝑧 ∈ 𝑑 → (𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠)))
90	89	rexlimdv 3139	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → (∃𝑧 ∈ 𝑑 𝑡 = (𝑧 ∩ 𝑆) → 𝑡 ∈ 𝑠))
91	77, 90	biimtrid 243	. . . . . . . . . . . . . . . 16 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → (𝑡 ∈ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} → 𝑡 ∈ 𝑠))
92	91	ssrdv 3928	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ⊆ 𝑠)
93		vex 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑑 ∈ V
94	93	abrexex 7911	. . . . . . . . . . . . . . . 16 ⊢ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ V
95	94	elpw 4540	. . . . . . . . . . . . . . 15 ⊢ ({𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ 𝒫 𝑠 ↔ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ⊆ 𝑠)
96	92, 95	sylibr 235	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ 𝒫 𝑠)
97		abrexfi 9259	. . . . . . . . . . . . . . . 16 ⊢ (𝑑 ∈ Fin → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ Fin)
98	97	ad2antlr 733	. . . . . . . . . . . . . . 15 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ Fin)
99	98	3adant3 1138	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ Fin)
100	96, 99	elind 4136	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ (𝒫 𝑠 ∩ Fin))
101		dfss 3909	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ⊆ ∪ 𝑑 ↔ 𝑆 = (𝑆 ∩ ∪ 𝑑))
102	101	biimpi 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ⊆ ∪ 𝑑 → 𝑆 = (𝑆 ∩ ∪ 𝑑))
103		uniiun 4995	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑑 = ∪ 𝑧 ∈ 𝑑 𝑧
104	103	ineq2i 4153	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∩ ∪ 𝑑) = (𝑆 ∩ ∪ 𝑧 ∈ 𝑑 𝑧)
105		iunin2 5007	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑧 ∈ 𝑑 (𝑆 ∩ 𝑧) = (𝑆 ∩ ∪ 𝑧 ∈ 𝑑 𝑧)
106		incom 4145	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ∩ 𝑧) = (𝑧 ∩ 𝑆)
107	106	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑑 → (𝑆 ∩ 𝑧) = (𝑧 ∩ 𝑆))
108	107	iuneq2i 4950	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑧 ∈ 𝑑 (𝑆 ∩ 𝑧) = ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆)
109	104, 105, 108	3eqtr2i 2769	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∩ ∪ 𝑑) = ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆)
110	102, 109	eqtrdi 2791	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ⊆ ∪ 𝑑 → 𝑆 = ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆))
111	110	3ad2ant2 1140	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → 𝑆 = ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆))
112	18	ad2antrl 734	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → 𝑆 = ∪ (𝐽 ↾_t 𝑆))
113	112	3adant1 1136	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → 𝑆 = ∪ (𝐽 ↾_t 𝑆))
114		vex 3436	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
115	114	inex1 5252	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∩ 𝑆) ∈ V
116	115	dfiun2 4968	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆) = ∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)}
117	116	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → ∪ 𝑧 ∈ 𝑑 (𝑧 ∩ 𝑆) = ∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)})
118	111, 113, 117	3eqtr3d 2783	. . . . . . . . . . . . 13 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → ∪ (𝐽 ↾_t 𝑆) = ∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)})
119		unieq 4856	. . . . . . . . . . . . . 14 ⊢ (𝑡 = {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} → ∪ 𝑡 = ∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)})
120	119	rspceeqv 3590	. . . . . . . . . . . . 13 ⊢ (({𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)} ∈ (𝒫 𝑠 ∩ Fin) ∧ ∪ (𝐽 ↾_t 𝑆) = ∪ {𝑥 ∣ ∃𝑧 ∈ 𝑑 𝑥 = (𝑧 ∩ 𝑆)}) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)
121	100, 118, 120	syl2anc 590	. . . . . . . . . . . 12 ⊢ (((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) ∧ 𝑆 ⊆ ∪ 𝑑 ∧ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠)) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)
122	121	3exp 1125	. . . . . . . . . . 11 ⊢ ((𝑑 ∈ 𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∧ 𝑑 ∈ Fin) → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))
123	73, 122	sylbi 218	. . . . . . . . . 10 ⊢ (𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin) → (𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))
124	123	rexlimiv 3134	. . . . . . . . 9 ⊢ (∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑 → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡))
125	72, 124	syl6 35	. . . . . . . 8 ⊢ (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))
126	125	com3r 87	. . . . . . 7 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → (𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))
127	71, 126	mpd 15	. . . . . 6 ⊢ ((((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) ∧ 𝑆 = ∪ 𝑠) → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡))
128	127	ex 413	. . . . 5 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → (𝑆 = ∪ 𝑠 → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))
129	19, 128	sylbird 261	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → (∪ (𝐽 ↾_t 𝑆) = ∪ 𝑠 → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))
130	129	com23 86	. . 3 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → ((𝑆 ⊆ ∪ {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} → ∃𝑑 ∈ (𝒫 {𝑦 ∈ 𝐽 ∣ (𝑦 ∩ 𝑆) ∈ 𝑠} ∩ Fin)𝑆 ⊆ ∪ 𝑑) → (∪ (𝐽 ↾_t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))
131	15, 130	syld 47	. 2 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) ∧ 𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → (∪ (𝐽 ↾_t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))
132	131	ralrimdva 3140	1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (∀𝑐 ∈ 𝒫 𝐽(𝑆 ⊆ ∪ 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑆 ⊆ ∪ 𝑑) → ∀𝑠 ∈ 𝒫 (𝐽 ↾_t 𝑆)(∪ (𝐽 ↾_t 𝑆) = ∪ 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)∪ (𝐽 ↾_t 𝑆) = ∪ 𝑡)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∃wex 1786 ∈ wcel 2119 {cab 2718 ∀wral 3054 ∃wrex 3064 {crab 3392 Vcvv 3432 ∩ cin 3889 ⊆ wss 3890 𝒫 cpw 4536 ∪ cuni 4845 ∪ ciun 4928 (class class class)co 7363 Fincfn 8890 ↾_t crest 17381 Topctop 22883

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-1o 8402 df-en 8891 df-dom 8892 df-fin 8894 df-fi 9321 df-rest 17383 df-topgen 17404 df-top 22884 df-topon 22901 df-bases 22936

This theorem is referenced by: cmpsub 23390

W3C validator