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Theorem uncmp 21619
Description: The union of two compact sets is compact. (Contributed by Jeff Hankins, 30-Jan-2010.)
Hypothesis
Ref Expression
uncmp.1 𝑋 = 𝐽
Assertion
Ref Expression
uncmp (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)

Proof of Theorem uncmp
Dummy variables 𝑐 𝑑 𝑚 𝑛 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpll 757 . 2 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → 𝐽 ∈ Top)
2 simpll 757 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝐽 ∈ Top)
3 ssun1 3999 . . . . . . . . . 10 𝑆 ⊆ (𝑆𝑇)
4 sseq2 3846 . . . . . . . . . 10 (𝑋 = (𝑆𝑇) → (𝑆𝑋𝑆 ⊆ (𝑆𝑇)))
53, 4mpbiri 250 . . . . . . . . 9 (𝑋 = (𝑆𝑇) → 𝑆𝑋)
65ad2antlr 717 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑆𝑋)
7 uncmp.1 . . . . . . . . 9 𝑋 = 𝐽
87cmpsub 21616 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛)))
92, 6, 8syl2anc 579 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛)))
10 simprr 763 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑋 = 𝑐)
116, 10sseqtrd 3860 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑆 𝑐)
12 unieq 4681 . . . . . . . . . . . 12 (𝑚 = 𝑐 𝑚 = 𝑐)
1312sseq2d 3852 . . . . . . . . . . 11 (𝑚 = 𝑐 → (𝑆 𝑚𝑆 𝑐))
14 pweq 4382 . . . . . . . . . . . . 13 (𝑚 = 𝑐 → 𝒫 𝑚 = 𝒫 𝑐)
1514ineq1d 4036 . . . . . . . . . . . 12 (𝑚 = 𝑐 → (𝒫 𝑚 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
1615rexeqdv 3341 . . . . . . . . . . 11 (𝑚 = 𝑐 → (∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛 ↔ ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛))
1713, 16imbi12d 336 . . . . . . . . . 10 (𝑚 = 𝑐 → ((𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛) ↔ (𝑆 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛)))
1817rspcv 3507 . . . . . . . . 9 (𝑐 ∈ 𝒫 𝐽 → (∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛) → (𝑆 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛)))
1918ad2antrl 718 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → (∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛) → (𝑆 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛)))
2011, 19mpid 44 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → (∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛) → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛))
219, 20sylbid 232 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝐽t 𝑆) ∈ Comp → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛))
22 ssun2 4000 . . . . . . . . . 10 𝑇 ⊆ (𝑆𝑇)
23 sseq2 3846 . . . . . . . . . 10 (𝑋 = (𝑆𝑇) → (𝑇𝑋𝑇 ⊆ (𝑆𝑇)))
2422, 23mpbiri 250 . . . . . . . . 9 (𝑋 = (𝑆𝑇) → 𝑇𝑋)
2524ad2antlr 717 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑇𝑋)
267cmpsub 21616 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((𝐽t 𝑇) ∈ Comp ↔ ∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠)))
272, 25, 26syl2anc 579 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝐽t 𝑇) ∈ Comp ↔ ∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠)))
2825, 10sseqtrd 3860 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑇 𝑐)
29 unieq 4681 . . . . . . . . . . . 12 (𝑟 = 𝑐 𝑟 = 𝑐)
3029sseq2d 3852 . . . . . . . . . . 11 (𝑟 = 𝑐 → (𝑇 𝑟𝑇 𝑐))
31 pweq 4382 . . . . . . . . . . . . 13 (𝑟 = 𝑐 → 𝒫 𝑟 = 𝒫 𝑐)
3231ineq1d 4036 . . . . . . . . . . . 12 (𝑟 = 𝑐 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
3332rexeqdv 3341 . . . . . . . . . . 11 (𝑟 = 𝑐 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠))
3430, 33imbi12d 336 . . . . . . . . . 10 (𝑟 = 𝑐 → ((𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠) ↔ (𝑇 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠)))
3534rspcv 3507 . . . . . . . . 9 (𝑐 ∈ 𝒫 𝐽 → (∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠) → (𝑇 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠)))
3635ad2antrl 718 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → (∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠) → (𝑇 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠)))
3728, 36mpid 44 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → (∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠) → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠))
3827, 37sylbid 232 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝐽t 𝑇) ∈ Comp → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠))
39 reeanv 3293 . . . . . . 7 (∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)(𝑆 𝑛𝑇 𝑠) ↔ (∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠))
40 elin 4019 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑐𝑛 ∈ Fin))
4140simplbi 493 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ 𝒫 𝑐)
4241elpwid 4391 . . . . . . . . . . . . . . 15 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛𝑐)
43 elin 4019 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑠 ∈ 𝒫 𝑐𝑠 ∈ Fin))
4443simplbi 493 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ 𝒫 𝑐)
4544elpwid 4391 . . . . . . . . . . . . . . 15 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠𝑐)
4642, 45anim12i 606 . . . . . . . . . . . . . 14 ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛𝑐𝑠𝑐))
4746ad2antrl 718 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑐𝑠𝑐))
48 unss 4010 . . . . . . . . . . . . 13 ((𝑛𝑐𝑠𝑐) ↔ (𝑛𝑠) ⊆ 𝑐)
4947, 48sylib 210 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ⊆ 𝑐)
5040simprbi 492 . . . . . . . . . . . . . 14 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ Fin)
5143simprbi 492 . . . . . . . . . . . . . 14 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ Fin)
52 unfi 8517 . . . . . . . . . . . . . 14 ((𝑛 ∈ Fin ∧ 𝑠 ∈ Fin) → (𝑛𝑠) ∈ Fin)
5350, 51, 52syl2an 589 . . . . . . . . . . . . 13 ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛𝑠) ∈ Fin)
5453ad2antrl 718 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ∈ Fin)
5549, 54jca 507 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → ((𝑛𝑠) ⊆ 𝑐 ∧ (𝑛𝑠) ∈ Fin))
56 elin 4019 . . . . . . . . . . . 12 ((𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin) ↔ ((𝑛𝑠) ∈ 𝒫 𝑐 ∧ (𝑛𝑠) ∈ Fin))
57 vex 3401 . . . . . . . . . . . . . 14 𝑐 ∈ V
5857elpw2 5064 . . . . . . . . . . . . 13 ((𝑛𝑠) ∈ 𝒫 𝑐 ↔ (𝑛𝑠) ⊆ 𝑐)
5958anbi1i 617 . . . . . . . . . . . 12 (((𝑛𝑠) ∈ 𝒫 𝑐 ∧ (𝑛𝑠) ∈ Fin) ↔ ((𝑛𝑠) ⊆ 𝑐 ∧ (𝑛𝑠) ∈ Fin))
6056, 59bitr2i 268 . . . . . . . . . . 11 (((𝑛𝑠) ⊆ 𝑐 ∧ (𝑛𝑠) ∈ Fin) ↔ (𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin))
6155, 60sylib 210 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin))
62 simpllr 766 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 = (𝑆𝑇))
63 ssun3 4001 . . . . . . . . . . . . . . . 16 (𝑆 𝑛𝑆 ⊆ ( 𝑛 𝑠))
64 ssun4 4002 . . . . . . . . . . . . . . . 16 (𝑇 𝑠𝑇 ⊆ ( 𝑛 𝑠))
6563, 64anim12i 606 . . . . . . . . . . . . . . 15 ((𝑆 𝑛𝑇 𝑠) → (𝑆 ⊆ ( 𝑛 𝑠) ∧ 𝑇 ⊆ ( 𝑛 𝑠)))
6665ad2antll 719 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑆 ⊆ ( 𝑛 𝑠) ∧ 𝑇 ⊆ ( 𝑛 𝑠)))
67 unss 4010 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ( 𝑛 𝑠) ∧ 𝑇 ⊆ ( 𝑛 𝑠)) ↔ (𝑆𝑇) ⊆ ( 𝑛 𝑠))
6866, 67sylib 210 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑆𝑇) ⊆ ( 𝑛 𝑠))
6962, 68eqsstrd 3858 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 ⊆ ( 𝑛 𝑠))
70 uniun 4694 . . . . . . . . . . . 12 (𝑛𝑠) = ( 𝑛 𝑠)
7169, 70syl6sseqr 3871 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 (𝑛𝑠))
72 elpwi 4389 . . . . . . . . . . . . . . 15 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
7372adantr 474 . . . . . . . . . . . . . 14 ((𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐) → 𝑐𝐽)
7473ad2antlr 717 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑐𝐽)
7549, 74sstrd 3831 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ⊆ 𝐽)
76 uniss 4696 . . . . . . . . . . . . 13 ((𝑛𝑠) ⊆ 𝐽 (𝑛𝑠) ⊆ 𝐽)
7776, 7syl6sseqr 3871 . . . . . . . . . . . 12 ((𝑛𝑠) ⊆ 𝐽 (𝑛𝑠) ⊆ 𝑋)
7875, 77syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ⊆ 𝑋)
7971, 78eqssd 3838 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 = (𝑛𝑠))
80 unieq 4681 . . . . . . . . . . 11 (𝑑 = (𝑛𝑠) → 𝑑 = (𝑛𝑠))
8180rspceeqv 3529 . . . . . . . . . 10 (((𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = (𝑛𝑠)) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
8261, 79, 81syl2anc 579 . . . . . . . . 9 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
8382exp32 413 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → ((𝑆 𝑛𝑇 𝑠) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
8483rexlimdvv 3220 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → (∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)(𝑆 𝑛𝑇 𝑠) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
8539, 84syl5bir 235 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛 ∧ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
8621, 38, 85syl2and 601 . . . . 5 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → (((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
8786impancom 445 . . . 4 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → ((𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
8887expd 406 . . 3 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
8988ralrimiv 3147 . 2 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
907iscmp 21604 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
911, 89, 90sylanbrc 578 1 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  cun 3790  cin 3791  wss 3792  𝒫 cpw 4379   cuni 4673  (class class class)co 6924  Fincfn 8243  t crest 16471  Topctop 21109  Compccmp 21602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-er 8028  df-en 8244  df-dom 8245  df-fin 8247  df-fi 8607  df-rest 16473  df-topgen 16494  df-top 21110  df-topon 21127  df-bases 21162  df-cmp 21603
This theorem is referenced by:  fiuncmp  21620
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