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Theorem cmpcld 21485
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 4322 . . . 4 (𝑠 ∈ 𝒫 𝐽𝑠𝐽)
2 simp1l 1254 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ∈ Comp)
3 simp2 1167 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠𝐽)
4 eqid 2765 . . . . . . . . . . . 12 𝐽 = 𝐽
54cldopn 21115 . . . . . . . . . . 11 (𝑆 ∈ (Clsd‘𝐽) → ( 𝐽𝑆) ∈ 𝐽)
65adantl 473 . . . . . . . . . 10 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ( 𝐽𝑆) ∈ 𝐽)
763ad2ant1 1163 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝐽𝑆) ∈ 𝐽)
87snssd 4494 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} ⊆ 𝐽)
93, 8unssd 3951 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑠 ∪ {( 𝐽𝑆)}) ⊆ 𝐽)
10 simp3 1168 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝑠)
11 uniss 4617 . . . . . . . . . . . . . 14 (𝑠𝐽 𝑠 𝐽)
12113ad2ant2 1164 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠 𝐽)
1310, 12sstrd 3771 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝐽)
14 undif 4209 . . . . . . . . . . . 12 (𝑆 𝐽 ↔ (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
1513, 14sylib 209 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
16 unss1 3944 . . . . . . . . . . . 12 (𝑆 𝑠 → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
17163ad2ant3 1165 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
1815, 17eqsstr3d 3800 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
19 difss 3899 . . . . . . . . . . 11 ( 𝐽𝑆) ⊆ 𝐽
20 unss 3949 . . . . . . . . . . 11 (( 𝑠 𝐽 ∧ ( 𝐽𝑆) ⊆ 𝐽) ↔ ( 𝑠 ∪ ( 𝐽𝑆)) ⊆ 𝐽)
2112, 19, 20sylanblc 583 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 ∪ ( 𝐽𝑆)) ⊆ 𝐽)
2218, 21eqssd 3778 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 ∪ ( 𝐽𝑆)))
23 uniexg 7153 . . . . . . . . . . . . 13 (𝐽 ∈ Comp → 𝐽 ∈ V)
2423ad2antrr 717 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽) → 𝐽 ∈ V)
25243adant3 1162 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ∈ V)
26 difexg 4969 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽𝑆) ∈ V)
27 unisng 4609 . . . . . . . . . . 11 (( 𝐽𝑆) ∈ V → {( 𝐽𝑆)} = ( 𝐽𝑆))
2825, 26, 273syl 18 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} = ( 𝐽𝑆))
2928uneq2d 3929 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 {( 𝐽𝑆)}) = ( 𝑠 ∪ ( 𝐽𝑆)))
3022, 29eqtr4d 2802 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 {( 𝐽𝑆)}))
31 uniun 4615 . . . . . . . 8 (𝑠 ∪ {( 𝐽𝑆)}) = ( 𝑠 {( 𝐽𝑆)})
3230, 31syl6eqr 2817 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = (𝑠 ∪ {( 𝐽𝑆)}))
334cmpcov 21472 . . . . . . 7 ((𝐽 ∈ Comp ∧ (𝑠 ∪ {( 𝐽𝑆)}) ⊆ 𝐽 𝐽 = (𝑠 ∪ {( 𝐽𝑆)})) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢)
342, 9, 32, 33syl3anc 1490 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ∃𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) 𝐽 = 𝑢)
35 elfpw 8475 . . . . . . . 8 (𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) ↔ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin))
36 simp2l 1256 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}))
37 uncom 3919 . . . . . . . . . . . 12 (𝑠 ∪ {( 𝐽𝑆)}) = ({( 𝐽𝑆)} ∪ 𝑠)
3836, 37syl6sseq 3811 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ ({( 𝐽𝑆)} ∪ 𝑠))
39 ssundif 4212 . . . . . . . . . . 11 (𝑢 ⊆ ({( 𝐽𝑆)} ∪ 𝑠) ↔ (𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠)
4038, 39sylib 209 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠)
41 diffi 8399 . . . . . . . . . . . 12 (𝑢 ∈ Fin → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
4241ad2antll 720 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin)) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
43423adant3 1162 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin)
44 elfpw 8475 . . . . . . . . . 10 ((𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ↔ ((𝑢 ∖ {( 𝐽𝑆)}) ⊆ 𝑠 ∧ (𝑢 ∖ {( 𝐽𝑆)}) ∈ Fin))
4540, 43, 44sylanbrc 578 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin))
46103ad2ant1 1163 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑠)
47123ad2ant1 1163 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝐽)
48 simp3 1168 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝐽 = 𝑢)
4947, 48sseqtrd 3801 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝑢)
5046, 49sstrd 3771 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑢)
5150sselda 3761 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → 𝑣 𝑢)
52 eluni 4597 . . . . . . . . . . . . . 14 (𝑣 𝑢 ↔ ∃𝑤(𝑣𝑤𝑤𝑢))
5351, 52sylib 209 . . . . . . . . . . . . 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 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 3896 . . . . . . . . . . . . . . . . . . . . . 22 (𝑣𝑆 → ¬ 𝑣 ∈ ( 𝐽𝑆))
5958ad2antlr 718 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) ∧ 𝑣𝑤) → ¬ 𝑣 ∈ ( 𝐽𝑆))
60 eleq2 2833 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑤 = ( 𝐽𝑆) → (𝑣𝑤𝑣 ∈ ( 𝐽𝑆)))
6160biimpd 220 . . . . . . . . . . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . . . . . . . . . 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 401 . . . . . . . . . . . . . . . . . . 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 4350 . . . . . . . . . . . . . . . . . . 19 (𝑤 ∈ {( 𝐽𝑆)} ↔ 𝑤 = ( 𝐽𝑆))
6968notbii 311 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ {( 𝐽𝑆)} ↔ ¬ 𝑤 = ( 𝐽𝑆))
7067, 69syl6ibr 243 . . . . . . . . . . . . . . . . 17 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → ¬ 𝑤 ∈ {( 𝐽𝑆)}))
7157, 70jcad 508 . . . . . . . . . . . . . . . 16 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → (𝑤𝑢 ∧ ¬ 𝑤 ∈ {( 𝐽𝑆)})))
72 eldif 3742 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}) ↔ (𝑤𝑢 ∧ ¬ 𝑤 ∈ {( 𝐽𝑆)}))
7371, 72syl6ibr 243 . . . . . . . . . . . . . . 15 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → 𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)})))
7455, 73jcad 508 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → ((𝑣𝑤𝑤𝑢) → (𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
7574eximdv 2012 . . . . . . . . . . . . 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 401 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑣𝑆 → ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)}))))
78 eluni 4597 . . . . . . . . . . 11 (𝑣 (𝑢 ∖ {( 𝐽𝑆)}) ↔ ∃𝑤(𝑣𝑤𝑤 ∈ (𝑢 ∖ {( 𝐽𝑆)})))
7977, 78syl6ibr 243 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → (𝑣𝑆𝑣 (𝑢 ∖ {( 𝐽𝑆)})))
8079ssrdv 3767 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 (𝑢 ∖ {( 𝐽𝑆)}))
81 unieq 4602 . . . . . . . . . . 11 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → 𝑡 = (𝑢 ∖ {( 𝐽𝑆)}))
8281sseq2d 3793 . . . . . . . . . 10 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → (𝑆 𝑡𝑆 (𝑢 ∖ {( 𝐽𝑆)})))
8382rspcev 3461 . . . . . . . . 9 (((𝑢 ∖ {( 𝐽𝑆)}) ∈ (𝒫 𝑠 ∩ Fin) ∧ 𝑆 (𝑢 ∖ {( 𝐽𝑆)})) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8445, 80, 83syl2anc 579 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8535, 84syl3an2b 1523 . . . . . . 7 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ 𝑢 ∈ (𝒫 (𝑠 ∪ {( 𝐽𝑆)}) ∩ Fin) ∧ 𝐽 = 𝑢) → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)
8685rexlimdv3a 3180 . . . . . 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 1148 . . . 4 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠𝐽 → (𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
891, 88syl5bi 233 . . 3 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝑠 ∈ 𝒫 𝐽 → (𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9089ralrimiv 3112 . 2 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡))
91 cmptop 21478 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
924cldss 21113 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
934cmpsub 21483 . . 3 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9491, 92, 93syl2an 589 . 2 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ((𝐽t 𝑆) ∈ Comp ↔ ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡)))
9590, 94mpbird 248 1 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → (𝐽t 𝑆) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cdif 3729  cun 3730  cin 3731  wss 3732  𝒫 cpw 4315  {csn 4334   cuni 4594  cfv 6068  (class class class)co 6842  Fincfn 8160  t crest 16349  Topctop 20977  Clsdccld 21100  Compccmp 21469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-fin 8164  df-fi 8524  df-rest 16351  df-topgen 16372  df-top 20978  df-topon 20995  df-bases 21030  df-cld 21103  df-cmp 21470
This theorem is referenced by:  hausllycmp  21577  cldllycmp  21578  txkgen  21735  cmphaushmeo  21883  cnheiborlem  23032  cmpcmet  23396  stoweidlem28  40814  stoweidlem50  40836  stoweidlem57  40843
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