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Theorem cmpcld 23128
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 velpw 4608 . . . 4 (𝑠 ∈ 𝒫 𝐽𝑠𝐽)
2 simp1l 1195 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ∈ Comp)
3 simp2 1135 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠𝐽)
4 eqid 2730 . . . . . . . . . . . 12 𝐽 = 𝐽
54cldopn 22757 . . . . . . . . . . 11 (𝑆 ∈ (Clsd‘𝐽) → ( 𝐽𝑆) ∈ 𝐽)
65adantl 480 . . . . . . . . . 10 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽𝑆) ∈ 𝐽)
763ad2ant1 1131 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝐽𝑆) ∈ 𝐽)
87snssd 4813 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} ⊆ 𝐽)
93, 8unssd 4187 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑠 ∪ {( 𝐽𝑆)}) ⊆ 𝐽)
10 simp3 1136 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝑠)
11 uniss 4917 . . . . . . . . . . . . . 14 (𝑠𝐽 𝑠 𝐽)
12113ad2ant2 1132 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠 𝐽)
1310, 12sstrd 3993 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝐽)
14 undif 4482 . . . . . . . . . . . 12 (𝑆 𝐽 ↔ (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
1513, 14sylib 217 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
16 unss1 4180 . . . . . . . . . . . 12 (𝑆 𝑠 → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
17163ad2ant3 1133 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
1815, 17eqsstrrd 4022 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
19 difss 4132 . . . . . . . . . . 11 ( 𝐽𝑆) ⊆ 𝐽
20 unss 4185 . . . . . . . . . . 11 (( 𝑠 𝐽 ∧ ( 𝐽𝑆) ⊆ 𝐽) ↔ ( 𝑠 ∪ ( 𝐽𝑆)) ⊆ 𝐽)
2112, 19, 20sylanblc 587 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 ∪ ( 𝐽𝑆)) ⊆ 𝐽)
2218, 21eqssd 4000 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 ∪ ( 𝐽𝑆)))
23 uniexg 7734 . . . . . . . . . . . . 13 (𝐽 ∈ Comp → 𝐽 ∈ V)
2423ad2antrr 722 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽) → 𝐽 ∈ V)
25243adant3 1130 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ∈ V)
26 difexg 5328 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽𝑆) ∈ V)
27 unisng 4930 . . . . . . . . . . 11 (( 𝐽𝑆) ∈ V → {( 𝐽𝑆)} = ( 𝐽𝑆))
2825, 26, 273syl 18 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} = ( 𝐽𝑆))
2928uneq2d 4164 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 {( 𝐽𝑆)}) = ( 𝑠 ∪ ( 𝐽𝑆)))
3022, 29eqtr4d 2773 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 {( 𝐽𝑆)}))
31 uniun 4935 . . . . . . . 8 (𝑠 ∪ {( 𝐽𝑆)}) = ( 𝑠 {( 𝐽𝑆)})
3230, 31eqtr4di 2788 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = (𝑠 ∪ {( 𝐽𝑆)}))
334cmpcov 23115 . . . . . . 7 ((𝐽 ∈ Comp ∧ (𝑠 ∪ {( 𝐽𝑆)}) ⊆ 𝐽 𝐽 = (𝑠 ∪ {( 𝐽𝑆)})) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢)
342, 9, 32, 33syl3anc 1369 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢)
35 elfpw 9358 . . . . . . . 8 (𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) ↔ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin))
36 simp2l 1197 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}))
37 uncom 4154 . . . . . . . . . . . 12 (𝑠 ∪ {( 𝐽𝑆)}) = ({( 𝐽𝑆)} ∪ 𝑠)
3836, 37sseqtrdi 4033 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ ({( 𝐽𝑆)} ∪ 𝑠))
39 ssundif 4488 . . . . . . . . . . 11 (𝑢 ⊆ ({( 𝐽𝑆)} ∪ 𝑠) ↔ (𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠)
4038, 39sylib 217 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠)
41 diffi 9183 . . . . . . . . . . . 12 (𝑢 ∈ Fin → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
4241ad2antll 725 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin)) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
43423adant3 1130 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
44 elfpw 9358 . . . . . . . . . 10 ((𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ↔ ((𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠 ∧ (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin))
4540, 43, 44sylanbrc 581 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin))
46103ad2ant1 1131 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑠)
47123ad2ant1 1131 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝐽)
48 simp3 1136 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝐽 = 𝑢)
4947, 48sseqtrd 4023 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝑢)
5046, 49sstrd 3993 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑢)
5150sselda 3983 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → 𝑣 𝑢)
52 eluni 4912 . . . . . . . . . . . . . 14 (𝑣 𝑢 ↔ ∃𝑤(𝑣𝑤𝑤𝑢))
5351, 52sylib 217 . . . . . . . . . . . . 13 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ∃𝑤(𝑣𝑤𝑤𝑢))
54 simpl 481 . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . 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 4129 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑆 → ¬ 𝑣 ∈ ( 𝐽𝑆))
5958ad2antlr 723 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) ∧ 𝑣𝑤) → ¬ 𝑣 ∈ ( 𝐽𝑆))
60 eleq2 2820 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = ( 𝐽𝑆) → (𝑣𝑤𝑣 ∈ ( 𝐽𝑆)))
6160biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) ∧ 𝑣𝑤) → (𝑤 = ( 𝐽𝑆) → 𝑣 ∈ ( 𝐽𝑆)))
6559, 64mtod 197 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) ∧ 𝑣𝑤) → ¬ 𝑤 = ( 𝐽𝑆))
6665ex 411 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → (𝑣𝑤 → ¬ 𝑤 = ( 𝐽𝑆)))
6766adantrd 490 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → ¬ 𝑤 = ( 𝐽𝑆)))
68 velsn 4645 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ {( 𝐽𝑆)} ↔ 𝑤 = ( 𝐽𝑆))
6968notbii 319 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ {( 𝐽𝑆)} ↔ ¬ 𝑤 = ( 𝐽𝑆))
7067, 69imbitrrdi 251 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → ¬ 𝑤 ∈ {( 𝐽𝑆)}))
7157, 70jcad 511 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → (𝑤𝑢 ∧ ¬ 𝑤 ∈ {( 𝐽𝑆)})))
72 eldif 3959 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}) ↔ (𝑤𝑢 ∧ ¬ 𝑤 ∈ {( 𝐽𝑆)}))
7371, 72imbitrrdi 251 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → 𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)})))
7455, 73jcad 511 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → (𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
7574eximdv 1918 . . . . . . . . . . . . 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 411 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑣𝑆 → ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
78 eluni 4912 . . . . . . . . . . 11 (𝑣 (𝑢 ∖ {( 𝐽𝑆)}) ↔ ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)})))
7977, 78imbitrrdi 251 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑣𝑆𝑣 (𝑢 ∖ {( 𝐽𝑆)})))
8079ssrdv 3989 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 (𝑢 ∖ {( 𝐽𝑆)}))
81 unieq 4920 . . . . . . . . . . 11 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → 𝑡 = (𝑢 ∖ {( 𝐽𝑆)}))
8281sseq2d 4015 . . . . . . . . . 10 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → (𝑆 𝑡𝑆 (𝑢 ∖ {( 𝐽𝑆)})))
8382rspcev 3613 . . . . . . . . 9 (((𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ∧ 𝑆 (𝑢 ∖ {( 𝐽𝑆)})) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8445, 80, 83syl2anc 582 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8535, 84syl3an2b 1402 . . . . . . 7 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ 𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) ∧ 𝐽 = 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8685rexlimdv3a 3157 . . . . . 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 1117 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠𝐽 → (𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
891, 88biimtrid 241 . . 3 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ∈ 𝒫 𝐽 → (𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9089ralrimiv 3143 . 2 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡))
91 cmptop 23121 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
924cldss 22755 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
934cmpsub 23126 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9491, 92, 93syl2an 594 . 2 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9590, 94mpbird 256 1 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽t 𝑆) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1085   = wceq 1539  wex 1779  wcel 2104  wral 3059  wrex 3068  Vcvv 3472  cdif 3946  cun 3947  cin 3948  wss 3949  𝒫 cpw 4603  {csn 4629   cuni 4909  cfv 6544  (class class class)co 7413  Fincfn 8943  t crest 17372  Topctop 22617  Clsdccld 22742  Compccmp 23112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-fin 8947  df-fi 9410  df-rest 17374  df-topgen 17395  df-top 22618  df-topon 22635  df-bases 22671  df-cld 22745  df-cmp 23113
This theorem is referenced by:  hausllycmp  23220  cldllycmp  23221  txkgen  23378  cmphaushmeo  23526  cnheiborlem  24702  cmpcmet  25069  stoweidlem28  45044  stoweidlem50  45066  stoweidlem57  45073
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