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Theorem uncmp 21939
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 763 . 2 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → 𝐽 ∈ Top)
2 simpll 763 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝐽 ∈ Top)
3 ssun1 4145 . . . . . . . . . 10 𝑆 ⊆ (𝑆𝑇)
4 sseq2 3990 . . . . . . . . . 10 (𝑋 = (𝑆𝑇) → (𝑆𝑋𝑆 ⊆ (𝑆𝑇)))
53, 4mpbiri 259 . . . . . . . . 9 (𝑋 = (𝑆𝑇) → 𝑆𝑋)
65ad2antlr 723 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑆𝑋)
7 uncmp.1 . . . . . . . . 9 𝑋 = 𝐽
87cmpsub 21936 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛)))
92, 6, 8syl2anc 584 . . . . . . 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 769 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑋 = 𝑐)
116, 10sseqtrd 4004 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑆 𝑐)
12 unieq 4838 . . . . . . . . . . . 12 (𝑚 = 𝑐 𝑚 = 𝑐)
1312sseq2d 3996 . . . . . . . . . . 11 (𝑚 = 𝑐 → (𝑆 𝑚𝑆 𝑐))
14 pweq 4538 . . . . . . . . . . . . 13 (𝑚 = 𝑐 → 𝒫 𝑚 = 𝒫 𝑐)
1514ineq1d 4185 . . . . . . . . . . . 12 (𝑚 = 𝑐 → (𝒫 𝑚 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
1615rexeqdv 3414 . . . . . . . . . . 11 (𝑚 = 𝑐 → (∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛 ↔ ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛))
1713, 16imbi12d 346 . . . . . . . . . 10 (𝑚 = 𝑐 → ((𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛) ↔ (𝑆 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛)))
1817rspcv 3615 . . . . . . . . 9 (𝑐 ∈ 𝒫 𝐽 → (∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛) → (𝑆 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛)))
1918ad2antrl 724 . . . . . . . 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 241 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝐽t 𝑆) ∈ Comp → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛))
22 ssun2 4146 . . . . . . . . . 10 𝑇 ⊆ (𝑆𝑇)
23 sseq2 3990 . . . . . . . . . 10 (𝑋 = (𝑆𝑇) → (𝑇𝑋𝑇 ⊆ (𝑆𝑇)))
2422, 23mpbiri 259 . . . . . . . . 9 (𝑋 = (𝑆𝑇) → 𝑇𝑋)
2524ad2antlr 723 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑇𝑋)
267cmpsub 21936 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((𝐽t 𝑇) ∈ Comp ↔ ∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠)))
272, 25, 26syl2anc 584 . . . . . . 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 4004 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑇 𝑐)
29 unieq 4838 . . . . . . . . . . . 12 (𝑟 = 𝑐 𝑟 = 𝑐)
3029sseq2d 3996 . . . . . . . . . . 11 (𝑟 = 𝑐 → (𝑇 𝑟𝑇 𝑐))
31 pweq 4538 . . . . . . . . . . . . 13 (𝑟 = 𝑐 → 𝒫 𝑟 = 𝒫 𝑐)
3231ineq1d 4185 . . . . . . . . . . . 12 (𝑟 = 𝑐 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
3332rexeqdv 3414 . . . . . . . . . . 11 (𝑟 = 𝑐 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠))
3430, 33imbi12d 346 . . . . . . . . . 10 (𝑟 = 𝑐 → ((𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠) ↔ (𝑇 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠)))
3534rspcv 3615 . . . . . . . . 9 (𝑐 ∈ 𝒫 𝐽 → (∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠) → (𝑇 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠)))
3635ad2antrl 724 . . . . . . . 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 241 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝐽t 𝑇) ∈ Comp → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠))
39 reeanv 3365 . . . . . . 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 elinel1 4169 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ 𝒫 𝑐)
4140elpwid 4549 . . . . . . . . . . . . . . 15 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛𝑐)
42 elinel1 4169 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ 𝒫 𝑐)
4342elpwid 4549 . . . . . . . . . . . . . . 15 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠𝑐)
4441, 43anim12i 612 . . . . . . . . . . . . . 14 ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛𝑐𝑠𝑐))
4544ad2antrl 724 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑐𝑠𝑐))
46 unss 4157 . . . . . . . . . . . . 13 ((𝑛𝑐𝑠𝑐) ↔ (𝑛𝑠) ⊆ 𝑐)
4745, 46sylib 219 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ⊆ 𝑐)
48 elinel2 4170 . . . . . . . . . . . . . 14 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ Fin)
49 elinel2 4170 . . . . . . . . . . . . . 14 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ Fin)
50 unfi 8773 . . . . . . . . . . . . . 14 ((𝑛 ∈ Fin ∧ 𝑠 ∈ Fin) → (𝑛𝑠) ∈ Fin)
5148, 49, 50syl2an 595 . . . . . . . . . . . . 13 ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛𝑠) ∈ Fin)
5251ad2antrl 724 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ∈ Fin)
5347, 52jca 512 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → ((𝑛𝑠) ⊆ 𝑐 ∧ (𝑛𝑠) ∈ Fin))
54 elin 4166 . . . . . . . . . . . 12 ((𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin) ↔ ((𝑛𝑠) ∈ 𝒫 𝑐 ∧ (𝑛𝑠) ∈ Fin))
55 vex 3495 . . . . . . . . . . . . . 14 𝑐 ∈ V
5655elpw2 5239 . . . . . . . . . . . . 13 ((𝑛𝑠) ∈ 𝒫 𝑐 ↔ (𝑛𝑠) ⊆ 𝑐)
5756anbi1i 623 . . . . . . . . . . . 12 (((𝑛𝑠) ∈ 𝒫 𝑐 ∧ (𝑛𝑠) ∈ Fin) ↔ ((𝑛𝑠) ⊆ 𝑐 ∧ (𝑛𝑠) ∈ Fin))
5854, 57bitr2i 277 . . . . . . . . . . 11 (((𝑛𝑠) ⊆ 𝑐 ∧ (𝑛𝑠) ∈ Fin) ↔ (𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin))
5953, 58sylib 219 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin))
60 simpllr 772 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 = (𝑆𝑇))
61 ssun3 4147 . . . . . . . . . . . . . . . 16 (𝑆 𝑛𝑆 ⊆ ( 𝑛 𝑠))
62 ssun4 4148 . . . . . . . . . . . . . . . 16 (𝑇 𝑠𝑇 ⊆ ( 𝑛 𝑠))
6361, 62anim12i 612 . . . . . . . . . . . . . . 15 ((𝑆 𝑛𝑇 𝑠) → (𝑆 ⊆ ( 𝑛 𝑠) ∧ 𝑇 ⊆ ( 𝑛 𝑠)))
6463ad2antll 725 . . . . . . . . . . . . . 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)))
65 unss 4157 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ( 𝑛 𝑠) ∧ 𝑇 ⊆ ( 𝑛 𝑠)) ↔ (𝑆𝑇) ⊆ ( 𝑛 𝑠))
6664, 65sylib 219 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑆𝑇) ⊆ ( 𝑛 𝑠))
6760, 66eqsstrd 4002 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 ⊆ ( 𝑛 𝑠))
68 uniun 4849 . . . . . . . . . . . 12 (𝑛𝑠) = ( 𝑛 𝑠)
6967, 68sseqtrrdi 4015 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 (𝑛𝑠))
70 elpwi 4547 . . . . . . . . . . . . . . 15 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
7170adantr 481 . . . . . . . . . . . . . 14 ((𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐) → 𝑐𝐽)
7271ad2antlr 723 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑐𝐽)
7347, 72sstrd 3974 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ⊆ 𝐽)
74 uniss 4851 . . . . . . . . . . . . 13 ((𝑛𝑠) ⊆ 𝐽 (𝑛𝑠) ⊆ 𝐽)
7574, 7sseqtrrdi 4015 . . . . . . . . . . . 12 ((𝑛𝑠) ⊆ 𝐽 (𝑛𝑠) ⊆ 𝑋)
7673, 75syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ⊆ 𝑋)
7769, 76eqssd 3981 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 = (𝑛𝑠))
78 unieq 4838 . . . . . . . . . . 11 (𝑑 = (𝑛𝑠) → 𝑑 = (𝑛𝑠))
7978rspceeqv 3635 . . . . . . . . . 10 (((𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = (𝑛𝑠)) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
8059, 77, 79syl2anc 584 . . . . . . . . 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)
8180exp32 421 . . . . . . . 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)))
8281rexlimdvv 3290 . . . . . . 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))
8339, 82syl5bir 244 . . . . . 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))
8421, 38, 83syl2and 607 . . . . 5 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → (((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
8584impancom 452 . . . 4 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → ((𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
8685expd 416 . . 3 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
8786ralrimiv 3178 . 2 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
887iscmp 21924 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
891, 87, 88sylanbrc 583 1 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cun 3931  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830  (class class class)co 7145  Fincfn 8497  t crest 16682  Topctop 21429  Compccmp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  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-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cmp 21923
This theorem is referenced by:  fiuncmp  21940
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