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Theorem uncmp 21016
 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 786 . 2 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → 𝐽 ∈ Top)
2 simpll 786 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝐽 ∈ Top)
3 ssun1 3738 . . . . . . . . . 10 𝑆 ⊆ (𝑆𝑇)
4 sseq2 3590 . . . . . . . . . 10 (𝑋 = (𝑆𝑇) → (𝑆𝑋𝑆 ⊆ (𝑆𝑇)))
53, 4mpbiri 247 . . . . . . . . 9 (𝑋 = (𝑆𝑇) → 𝑆𝑋)
65ad2antlr 759 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑆𝑋)
7 uncmp.1 . . . . . . . . 9 𝑋 = 𝐽
87cmpsub 21013 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑆𝑋) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛)))
92, 6, 8syl2anc 691 . . . . . . 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 792 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑋 = 𝑐)
116, 10sseqtrd 3604 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑆 𝑐)
12 unieq 4380 . . . . . . . . . . . 12 (𝑚 = 𝑐 𝑚 = 𝑐)
1312sseq2d 3596 . . . . . . . . . . 11 (𝑚 = 𝑐 → (𝑆 𝑚𝑆 𝑐))
14 pweq 4111 . . . . . . . . . . . . 13 (𝑚 = 𝑐 → 𝒫 𝑚 = 𝒫 𝑐)
1514ineq1d 3775 . . . . . . . . . . . 12 (𝑚 = 𝑐 → (𝒫 𝑚 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
1615rexeqdv 3122 . . . . . . . . . . 11 (𝑚 = 𝑐 → (∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛 ↔ ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛))
1713, 16imbi12d 333 . . . . . . . . . 10 (𝑚 = 𝑐 → ((𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛) ↔ (𝑆 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛)))
1817rspcv 3278 . . . . . . . . 9 (𝑐 ∈ 𝒫 𝐽 → (∀𝑚 ∈ 𝒫 𝐽(𝑆 𝑚 → ∃𝑛 ∈ (𝒫 𝑚 ∩ Fin)𝑆 𝑛) → (𝑆 𝑐 → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛)))
1918ad2antrl 760 . . . . . . . 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 43 . . . . . . 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 229 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝐽t 𝑆) ∈ Comp → ∃𝑛 ∈ (𝒫 𝑐 ∩ Fin)𝑆 𝑛))
22 ssun2 3739 . . . . . . . . . 10 𝑇 ⊆ (𝑆𝑇)
23 sseq2 3590 . . . . . . . . . 10 (𝑋 = (𝑆𝑇) → (𝑇𝑋𝑇 ⊆ (𝑆𝑇)))
2422, 23mpbiri 247 . . . . . . . . 9 (𝑋 = (𝑆𝑇) → 𝑇𝑋)
2524ad2antlr 759 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑇𝑋)
267cmpsub 21013 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑇𝑋) → ((𝐽t 𝑇) ∈ Comp ↔ ∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠)))
272, 25, 26syl2anc 691 . . . . . . 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 3604 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → 𝑇 𝑐)
29 unieq 4380 . . . . . . . . . . . 12 (𝑟 = 𝑐 𝑟 = 𝑐)
3029sseq2d 3596 . . . . . . . . . . 11 (𝑟 = 𝑐 → (𝑇 𝑟𝑇 𝑐))
31 pweq 4111 . . . . . . . . . . . . 13 (𝑟 = 𝑐 → 𝒫 𝑟 = 𝒫 𝑐)
3231ineq1d 3775 . . . . . . . . . . . 12 (𝑟 = 𝑐 → (𝒫 𝑟 ∩ Fin) = (𝒫 𝑐 ∩ Fin))
3332rexeqdv 3122 . . . . . . . . . . 11 (𝑟 = 𝑐 → (∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠 ↔ ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠))
3430, 33imbi12d 333 . . . . . . . . . 10 (𝑟 = 𝑐 → ((𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠) ↔ (𝑇 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠)))
3534rspcv 3278 . . . . . . . . 9 (𝑐 ∈ 𝒫 𝐽 → (∀𝑟 ∈ 𝒫 𝐽(𝑇 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)𝑇 𝑠) → (𝑇 𝑐 → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠)))
3635ad2antrl 760 . . . . . . . 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 43 . . . . . . 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 229 . . . . . 6 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → ((𝐽t 𝑇) ∈ Comp → ∃𝑠 ∈ (𝒫 𝑐 ∩ Fin)𝑇 𝑠))
39 reeanv 3086 . . . . . . 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 3758 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑐𝑛 ∈ Fin))
4140simplbi 475 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ 𝒫 𝑐)
4241elpwid 4118 . . . . . . . . . . . . . . 15 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛𝑐)
43 elin 3758 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑠 ∈ 𝒫 𝑐𝑠 ∈ Fin))
4443simplbi 475 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ 𝒫 𝑐)
4544elpwid 4118 . . . . . . . . . . . . . . 15 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠𝑐)
4642, 45anim12i 588 . . . . . . . . . . . . . 14 ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛𝑐𝑠𝑐))
4746ad2antrl 760 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑐𝑠𝑐))
48 unss 3749 . . . . . . . . . . . . 13 ((𝑛𝑐𝑠𝑐) ↔ (𝑛𝑠) ⊆ 𝑐)
4947, 48sylib 207 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ⊆ 𝑐)
5040simprbi 479 . . . . . . . . . . . . . 14 (𝑛 ∈ (𝒫 𝑐 ∩ Fin) → 𝑛 ∈ Fin)
5143simprbi 479 . . . . . . . . . . . . . 14 (𝑠 ∈ (𝒫 𝑐 ∩ Fin) → 𝑠 ∈ Fin)
52 unfi 8112 . . . . . . . . . . . . . 14 ((𝑛 ∈ Fin ∧ 𝑠 ∈ Fin) → (𝑛𝑠) ∈ Fin)
5350, 51, 52syl2an 493 . . . . . . . . . . . . 13 ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) → (𝑛𝑠) ∈ Fin)
5453ad2antrl 760 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ∈ Fin)
5549, 54jca 553 . . . . . . . . . . 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 3758 . . . . . . . . . . . 12 ((𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin) ↔ ((𝑛𝑠) ∈ 𝒫 𝑐 ∧ (𝑛𝑠) ∈ Fin))
57 vex 3176 . . . . . . . . . . . . . 14 𝑐 ∈ V
5857elpw2 4755 . . . . . . . . . . . . 13 ((𝑛𝑠) ∈ 𝒫 𝑐 ↔ (𝑛𝑠) ⊆ 𝑐)
5958anbi1i 727 . . . . . . . . . . . 12 (((𝑛𝑠) ∈ 𝒫 𝑐 ∧ (𝑛𝑠) ∈ Fin) ↔ ((𝑛𝑠) ⊆ 𝑐 ∧ (𝑛𝑠) ∈ Fin))
6056, 59bitr2i 264 . . . . . . . . . . 11 (((𝑛𝑠) ⊆ 𝑐 ∧ (𝑛𝑠) ∈ Fin) ↔ (𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin))
6155, 60sylib 207 . . . . . . . . . 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 795 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 = (𝑆𝑇))
63 ssun3 3740 . . . . . . . . . . . . . . . 16 (𝑆 𝑛𝑆 ⊆ ( 𝑛 𝑠))
64 ssun4 3741 . . . . . . . . . . . . . . . 16 (𝑇 𝑠𝑇 ⊆ ( 𝑛 𝑠))
6563, 64anim12i 588 . . . . . . . . . . . . . . 15 ((𝑆 𝑛𝑇 𝑠) → (𝑆 ⊆ ( 𝑛 𝑠) ∧ 𝑇 ⊆ ( 𝑛 𝑠)))
6665ad2antll 761 . . . . . . . . . . . . . 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 3749 . . . . . . . . . . . . . 14 ((𝑆 ⊆ ( 𝑛 𝑠) ∧ 𝑇 ⊆ ( 𝑛 𝑠)) ↔ (𝑆𝑇) ⊆ ( 𝑛 𝑠))
6866, 67sylib 207 . . . . . . . . . . . . 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 3602 . . . . . . . . . . . 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 4392 . . . . . . . . . . . 12 (𝑛𝑠) = ( 𝑛 𝑠)
7169, 70syl6sseqr 3615 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 (𝑛𝑠))
72 elpwi 4117 . . . . . . . . . . . . . . 15 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
7372adantr 480 . . . . . . . . . . . . . 14 ((𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐) → 𝑐𝐽)
7473ad2antlr 759 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑐𝐽)
7549, 74sstrd 3578 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → (𝑛𝑠) ⊆ 𝐽)
76 uniss 4394 . . . . . . . . . . . . 13 ((𝑛𝑠) ⊆ 𝐽 (𝑛𝑠) ⊆ 𝐽)
7776, 7syl6sseqr 3615 . . . . . . . . . . . 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 3585 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) ∧ ((𝑛 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑠 ∈ (𝒫 𝑐 ∩ Fin)) ∧ (𝑆 𝑛𝑇 𝑠))) → 𝑋 = (𝑛𝑠))
80 unieq 4380 . . . . . . . . . . . 12 (𝑑 = (𝑛𝑠) → 𝑑 = (𝑛𝑠))
8180eqeq2d 2620 . . . . . . . . . . 11 (𝑑 = (𝑛𝑠) → (𝑋 = 𝑑𝑋 = (𝑛𝑠)))
8281rspcev 3282 . . . . . . . . . 10 (((𝑛𝑠) ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = (𝑛𝑠)) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
8361, 79, 82syl2anc 691 . . . . . . . . 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)
8483exp32 629 . . . . . . . 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)))
8584rexlimdvv 3019 . . . . . . 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))
8639, 85syl5bir 232 . . . . . 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))
8721, 38, 86syl2and 499 . . . . 5 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ (𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐)) → (((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
8887impancom 455 . . . 4 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → ((𝑐 ∈ 𝒫 𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
8988expd 451 . . 3 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
9089ralrimiv 2948 . 2 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑))
917iscmp 21001 . 2 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)))
921, 90, 91sylanbrc 695 1 (((𝐽 ∈ Top ∧ 𝑋 = (𝑆𝑇)) ∧ ((𝐽t 𝑆) ∈ Comp ∧ (𝐽t 𝑇) ∈ Comp)) → 𝐽 ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  (class class class)co 6549  Fincfn 7841   ↾t crest 15904  Topctop 20517  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000 This theorem is referenced by:  fiuncmp  21017
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