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Theorem cmpcld 21115
Description: A closed subset of a compact space is compact. (Contributed by Jeff Hankins, 29-Jun-2009.)
Assertion
Ref Expression
cmpcld ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽t 𝑆) ∈ Comp)

Proof of Theorem cmpcld
Dummy variables 𝑡 𝑠 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 selpw 4137 . . . 4 (𝑠 ∈ 𝒫 𝐽𝑠𝐽)
2 simp1l 1083 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ∈ Comp)
3 simp2 1060 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠𝐽)
4 eqid 2621 . . . . . . . . . . . 12 𝐽 = 𝐽
54cldopn 20745 . . . . . . . . . . 11 (𝑆 ∈ (Clsd‘𝐽) → ( 𝐽𝑆) ∈ 𝐽)
65adantl 482 . . . . . . . . . 10 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽𝑆) ∈ 𝐽)
763ad2ant1 1080 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝐽𝑆) ∈ 𝐽)
87snssd 4309 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} ⊆ 𝐽)
93, 8unssd 3767 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑠 ∪ {( 𝐽𝑆)}) ⊆ 𝐽)
10 simp3 1061 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝑠)
11 uniss 4424 . . . . . . . . . . . . . 14 (𝑠𝐽 𝑠 𝐽)
12113ad2ant2 1081 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠 𝐽)
1310, 12sstrd 3593 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝐽)
14 undif 4021 . . . . . . . . . . . 12 (𝑆 𝐽 ↔ (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
1513, 14sylib 208 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
16 unss1 3760 . . . . . . . . . . . 12 (𝑆 𝑠 → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
17163ad2ant3 1082 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
1815, 17eqsstr3d 3619 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
19 difss 3715 . . . . . . . . . . 11 ( 𝐽𝑆) ⊆ 𝐽
20 unss 3765 . . . . . . . . . . 11 (( 𝑠 𝐽 ∧ ( 𝐽𝑆) ⊆ 𝐽) ↔ ( 𝑠 ∪ ( 𝐽𝑆)) ⊆ 𝐽)
2112, 19, 20sylanblc 695 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 ∪ ( 𝐽𝑆)) ⊆ 𝐽)
2218, 21eqssd 3600 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 ∪ ( 𝐽𝑆)))
23 uniexg 6908 . . . . . . . . . . . . 13 (𝐽 ∈ Comp → 𝐽 ∈ V)
2423ad2antrr 761 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽) → 𝐽 ∈ V)
25243adant3 1079 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ∈ V)
26 difexg 4768 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽𝑆) ∈ V)
27 unisng 4418 . . . . . . . . . . 11 (( 𝐽𝑆) ∈ V → {( 𝐽𝑆)} = ( 𝐽𝑆))
2825, 26, 273syl 18 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} = ( 𝐽𝑆))
2928uneq2d 3745 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 {( 𝐽𝑆)}) = ( 𝑠 ∪ ( 𝐽𝑆)))
3022, 29eqtr4d 2658 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 {( 𝐽𝑆)}))
31 uniun 4422 . . . . . . . 8 (𝑠 ∪ {( 𝐽𝑆)}) = ( 𝑠 {( 𝐽𝑆)})
3230, 31syl6eqr 2673 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = (𝑠 ∪ {( 𝐽𝑆)}))
334cmpcov 21102 . . . . . . 7 ((𝐽 ∈ Comp ∧ (𝑠 ∪ {( 𝐽𝑆)}) ⊆ 𝐽 𝐽 = (𝑠 ∪ {( 𝐽𝑆)})) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢)
342, 9, 32, 33syl3anc 1323 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢)
35 elfpw 8212 . . . . . . . 8 (𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) ↔ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin))
36 simp2l 1085 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}))
37 uncom 3735 . . . . . . . . . . . 12 (𝑠 ∪ {( 𝐽𝑆)}) = ({( 𝐽𝑆)} ∪ 𝑠)
3836, 37syl6sseq 3630 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ ({( 𝐽𝑆)} ∪ 𝑠))
39 ssundif 4024 . . . . . . . . . . 11 (𝑢 ⊆ ({( 𝐽𝑆)} ∪ 𝑠) ↔ (𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠)
4038, 39sylib 208 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠)
41 diffi 8136 . . . . . . . . . . . 12 (𝑢 ∈ Fin → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
4241ad2antll 764 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin)) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
43423adant3 1079 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
44 elfpw 8212 . . . . . . . . . 10 ((𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ↔ ((𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠 ∧ (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin))
4540, 43, 44sylanbrc 697 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin))
46103ad2ant1 1080 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑠)
47123ad2ant1 1080 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝐽)
48 simp3 1061 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝐽 = 𝑢)
4947, 48sseqtrd 3620 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝑢)
5046, 49sstrd 3593 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑢)
5150sselda 3583 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → 𝑣 𝑢)
52 eluni 4405 . . . . . . . . . . . . . 14 (𝑣 𝑢 ↔ ∃𝑤(𝑣𝑤𝑤𝑢))
5351, 52sylib 208 . . . . . . . . . . . . 13 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ∃𝑤(𝑣𝑤𝑤𝑢))
54 simpl 473 . . . . . . . . . . . . . . . 16 ((𝑣𝑤𝑤𝑢) → 𝑣𝑤)
5554a1i 11 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → 𝑣𝑤))
56 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝑣𝑤𝑤𝑢) → 𝑤𝑢)
5756a1i 11 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → 𝑤𝑢))
58 elndif 3712 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑆 → ¬ 𝑣 ∈ ( 𝐽𝑆))
5958ad2antlr 762 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) ∧ 𝑣𝑤) → ¬ 𝑣 ∈ ( 𝐽𝑆))
60 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = ( 𝐽𝑆) → (𝑣𝑤𝑣 ∈ ( 𝐽𝑆)))
6160biimpd 219 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤 = ( 𝐽𝑆) → (𝑣𝑤𝑣 ∈ ( 𝐽𝑆)))
6261a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → (𝑤 = ( 𝐽𝑆) → (𝑣𝑤𝑣 ∈ ( 𝐽𝑆))))
6362com23 86 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → (𝑣𝑤 → (𝑤 = ( 𝐽𝑆) → 𝑣 ∈ ( 𝐽𝑆))))
6463imp 445 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) ∧ 𝑣𝑤) → (𝑤 = ( 𝐽𝑆) → 𝑣 ∈ ( 𝐽𝑆)))
6559, 64mtod 189 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) ∧ 𝑣𝑤) → ¬ 𝑤 = ( 𝐽𝑆))
6665ex 450 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → (𝑣𝑤 → ¬ 𝑤 = ( 𝐽𝑆)))
6766adantrd 484 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → ¬ 𝑤 = ( 𝐽𝑆)))
68 velsn 4164 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ {( 𝐽𝑆)} ↔ 𝑤 = ( 𝐽𝑆))
6968notbii 310 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ {( 𝐽𝑆)} ↔ ¬ 𝑤 = ( 𝐽𝑆))
7067, 69syl6ibr 242 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → ¬ 𝑤 ∈ {( 𝐽𝑆)}))
7157, 70jcad 555 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → (𝑤𝑢 ∧ ¬ 𝑤 ∈ {( 𝐽𝑆)})))
72 eldif 3565 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}) ↔ (𝑤𝑢 ∧ ¬ 𝑤 ∈ {( 𝐽𝑆)}))
7371, 72syl6ibr 242 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → 𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)})))
7455, 73jcad 555 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → (𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
7574eximdv 1843 . . . . . . . . . . . . 13 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → (∃𝑤(𝑣𝑤𝑤𝑢) → ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
7653, 75mpd 15 . . . . . . . . . . . 12 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)})))
7776ex 450 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑣𝑆 → ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
78 eluni 4405 . . . . . . . . . . 11 (𝑣 (𝑢 ∖ {( 𝐽𝑆)}) ↔ ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)})))
7977, 78syl6ibr 242 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑣𝑆𝑣 (𝑢 ∖ {( 𝐽𝑆)})))
8079ssrdv 3589 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 (𝑢 ∖ {( 𝐽𝑆)}))
81 unieq 4410 . . . . . . . . . . 11 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → 𝑡 = (𝑢 ∖ {( 𝐽𝑆)}))
8281sseq2d 3612 . . . . . . . . . 10 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → (𝑆 𝑡𝑆 (𝑢 ∖ {( 𝐽𝑆)})))
8382rspcev 3295 . . . . . . . . 9 (((𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ∧ 𝑆 (𝑢 ∖ {( 𝐽𝑆)})) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8445, 80, 83syl2anc 692 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8535, 84syl3an2b 1360 . . . . . . 7 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ 𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) ∧ 𝐽 = 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8685rexlimdv3a 3026 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡))
8734, 86mpd 15 . . . . 5 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
88873exp 1261 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠𝐽 → (𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
891, 88syl5bi 232 . . 3 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ∈ 𝒫 𝐽 → (𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9089ralrimiv 2959 . 2 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡))
91 cmptop 21108 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
924cldss 20743 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
934cmpsub 21113 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9491, 92, 93syl2an 494 . 2 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9590, 94mpbird 247 1 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽t 𝑆) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  𝒫 cpw 4130  {csn 4148   cuni 4402  cfv 5847  (class class class)co 6604  Fincfn 7899  t crest 16002  Topctop 20617  Clsdccld 20730  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cld 20733  df-cmp 21100
This theorem is referenced by:  hausllycmp  21207  cldllycmp  21208  txkgen  21365  cmphaushmeo  21513  cnheiborlem  22661  cmpcmet  23024  stoweidlem28  39549  stoweidlem50  39571  stoweidlem57  39578
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