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Theorem cmpcld 21403
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 4305 . . . 4 (𝑠 ∈ 𝒫 𝐽𝑠𝐽)
2 simp1l 1240 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ∈ Comp)
3 simp2 1132 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠𝐽)
4 eqid 2756 . . . . . . . . . . . 12 𝐽 = 𝐽
54cldopn 21033 . . . . . . . . . . 11 (𝑆 ∈ (Clsd‘𝐽) → ( 𝐽𝑆) ∈ 𝐽)
65adantl 473 . . . . . . . . . 10 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽𝑆) ∈ 𝐽)
763ad2ant1 1128 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝐽𝑆) ∈ 𝐽)
87snssd 4481 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} ⊆ 𝐽)
93, 8unssd 3928 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑠 ∪ {( 𝐽𝑆)}) ⊆ 𝐽)
10 simp3 1133 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝑠)
11 uniss 4606 . . . . . . . . . . . . . 14 (𝑠𝐽 𝑠 𝐽)
12113ad2ant2 1129 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠 𝐽)
1310, 12sstrd 3750 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝐽)
14 undif 4189 . . . . . . . . . . . 12 (𝑆 𝐽 ↔ (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
1513, 14sylib 208 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
16 unss1 3921 . . . . . . . . . . . 12 (𝑆 𝑠 → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
17163ad2ant3 1130 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
1815, 17eqsstr3d 3777 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
19 difss 3876 . . . . . . . . . . 11 ( 𝐽𝑆) ⊆ 𝐽
20 unss 3926 . . . . . . . . . . 11 (( 𝑠 𝐽 ∧ ( 𝐽𝑆) ⊆ 𝐽) ↔ ( 𝑠 ∪ ( 𝐽𝑆)) ⊆ 𝐽)
2112, 19, 20sylanblc 699 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 ∪ ( 𝐽𝑆)) ⊆ 𝐽)
2218, 21eqssd 3757 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 ∪ ( 𝐽𝑆)))
23 uniexg 7116 . . . . . . . . . . . . 13 (𝐽 ∈ Comp → 𝐽 ∈ V)
2423ad2antrr 764 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽) → 𝐽 ∈ V)
25243adant3 1127 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ∈ V)
26 difexg 4956 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽𝑆) ∈ V)
27 unisng 4600 . . . . . . . . . . 11 (( 𝐽𝑆) ∈ V → {( 𝐽𝑆)} = ( 𝐽𝑆))
2825, 26, 273syl 18 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} = ( 𝐽𝑆))
2928uneq2d 3906 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 {( 𝐽𝑆)}) = ( 𝑠 ∪ ( 𝐽𝑆)))
3022, 29eqtr4d 2793 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 {( 𝐽𝑆)}))
31 uniun 4604 . . . . . . . 8 (𝑠 ∪ {( 𝐽𝑆)}) = ( 𝑠 {( 𝐽𝑆)})
3230, 31syl6eqr 2808 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = (𝑠 ∪ {( 𝐽𝑆)}))
334cmpcov 21390 . . . . . . 7 ((𝐽 ∈ Comp ∧ (𝑠 ∪ {( 𝐽𝑆)}) ⊆ 𝐽 𝐽 = (𝑠 ∪ {( 𝐽𝑆)})) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢)
342, 9, 32, 33syl3anc 1477 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢)
35 elfpw 8429 . . . . . . . 8 (𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) ↔ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin))
36 simp2l 1242 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}))
37 uncom 3896 . . . . . . . . . . . 12 (𝑠 ∪ {( 𝐽𝑆)}) = ({( 𝐽𝑆)} ∪ 𝑠)
3836, 37syl6sseq 3788 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ ({( 𝐽𝑆)} ∪ 𝑠))
39 ssundif 4192 . . . . . . . . . . 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 8353 . . . . . . . . . . . 12 (𝑢 ∈ Fin → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
4241ad2antll 767 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin)) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
43423adant3 1127 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
44 elfpw 8429 . . . . . . . . . 10 ((𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ↔ ((𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠 ∧ (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin))
4540, 43, 44sylanbrc 701 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin))
46103ad2ant1 1128 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑠)
47123ad2ant1 1128 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝐽)
48 simp3 1133 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝐽 = 𝑢)
4947, 48sseqtrd 3778 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝑢)
5046, 49sstrd 3750 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑢)
5150sselda 3740 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → 𝑣 𝑢)
52 eluni 4587 . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 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 3873 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑆 → ¬ 𝑣 ∈ ( 𝐽𝑆))
5958ad2antlr 765 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) ∧ 𝑣𝑤) → ¬ 𝑣 ∈ ( 𝐽𝑆))
60 eleq2 2824 . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → (𝑣𝑤 → ¬ 𝑤 = ( 𝐽𝑆)))
6766adantrd 485 . . . . . . . . . . . . . . . . . 18 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → ¬ 𝑤 = ( 𝐽𝑆)))
68 velsn 4333 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ {( 𝐽𝑆)} ↔ 𝑤 = ( 𝐽𝑆))
6968notbii 309 . . . . . . . . . . . . . . . . . 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 556 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → (𝑤𝑢 ∧ ¬ 𝑤 ∈ {( 𝐽𝑆)})))
72 eldif 3721 . . . . . . . . . . . . . . . 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 556 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → (𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
7574eximdv 1991 . . . . . . . . . . . . 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 449 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑣𝑆 → ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
78 eluni 4587 . . . . . . . . . . 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 3746 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 (𝑢 ∖ {( 𝐽𝑆)}))
81 unieq 4592 . . . . . . . . . . 11 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → 𝑡 = (𝑢 ∖ {( 𝐽𝑆)}))
8281sseq2d 3770 . . . . . . . . . 10 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → (𝑆 𝑡𝑆 (𝑢 ∖ {( 𝐽𝑆)})))
8382rspcev 3445 . . . . . . . . 9 (((𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ∧ 𝑆 (𝑢 ∖ {( 𝐽𝑆)})) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8445, 80, 83syl2anc 696 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8535, 84syl3an2b 1511 . . . . . . 7 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ 𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) ∧ 𝐽 = 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8685rexlimdv3a 3167 . . . . . 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 1113 . . . 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 3099 . 2 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡))
91 cmptop 21396 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
924cldss 21031 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
934cmpsub 21401 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9491, 92, 93syl2an 495 . 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 383  w3a 1072   = wceq 1628  wex 1849  wcel 2135  wral 3046  wrex 3047  Vcvv 3336  cdif 3708  cun 3709  cin 3710  wss 3711  𝒫 cpw 4298  {csn 4317   cuni 4584  cfv 6045  (class class class)co 6809  Fincfn 8117  t crest 16279  Topctop 20896  Clsdccld 21018  Compccmp 21387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-pss 3727  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4585  df-int 4624  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-tr 4901  df-id 5170  df-eprel 5175  df-po 5183  df-so 5184  df-fr 5221  df-we 5223  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-pred 5837  df-ord 5883  df-on 5884  df-lim 5885  df-suc 5886  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-om 7227  df-1st 7329  df-2nd 7330  df-wrecs 7572  df-recs 7633  df-rdg 7671  df-1o 7725  df-oadd 7729  df-er 7907  df-en 8118  df-dom 8119  df-fin 8121  df-fi 8478  df-rest 16281  df-topgen 16302  df-top 20897  df-topon 20914  df-bases 20948  df-cld 21021  df-cmp 21388
This theorem is referenced by:  hausllycmp  21495  cldllycmp  21496  txkgen  21653  cmphaushmeo  21801  cnheiborlem  22950  cmpcmet  23312  stoweidlem28  40744  stoweidlem50  40766  stoweidlem57  40773
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