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Theorem cmpcld 21407
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 4309 . . . 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 2760 . . . . . . . . . . . 12 𝐽 = 𝐽
54cldopn 21037 . . . . . . . . . . 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 4485 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} ⊆ 𝐽)
93, 8unssd 3932 . . . . . . 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 4610 . . . . . . . . . . . . . 14 (𝑠𝐽 𝑠 𝐽)
12113ad2ant2 1129 . . . . . . . . . . . . 13 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑠 𝐽)
1310, 12sstrd 3754 . . . . . . . . . . . 12 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝑆 𝐽)
14 undif 4193 . . . . . . . . . . . 12 (𝑆 𝐽 ↔ (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
1513, 14sylib 208 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) = 𝐽)
16 unss1 3925 . . . . . . . . . . . 12 (𝑆 𝑠 → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
17163ad2ant3 1130 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → (𝑆 ∪ ( 𝐽𝑆)) ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
1815, 17eqsstr3d 3781 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 ⊆ ( 𝑠 ∪ ( 𝐽𝑆)))
19 difss 3880 . . . . . . . . . . 11 ( 𝐽𝑆) ⊆ 𝐽
20 unss 3930 . . . . . . . . . . 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 3761 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 ∪ ( 𝐽𝑆)))
23 uniexg 7120 . . . . . . . . . . . . 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 4960 . . . . . . . . . . 11 ( 𝐽 ∈ V → ( 𝐽𝑆) ∈ V)
27 unisng 4604 . . . . . . . . . . 11 (( 𝐽𝑆) ∈ V → {( 𝐽𝑆)} = ( 𝐽𝑆))
2825, 26, 273syl 18 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → {( 𝐽𝑆)} = ( 𝐽𝑆))
2928uneq2d 3910 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → ( 𝑠 {( 𝐽𝑆)}) = ( 𝑠 ∪ ( 𝐽𝑆)))
3022, 29eqtr4d 2797 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = ( 𝑠 {( 𝐽𝑆)}))
31 uniun 4608 . . . . . . . 8 (𝑠 ∪ {( 𝐽𝑆)}) = ( 𝑠 {( 𝐽𝑆)})
3230, 31syl6eqr 2812 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) → 𝐽 = (𝑠 ∪ {( 𝐽𝑆)}))
334cmpcov 21394 . . . . . . 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 8433 . . . . . . . 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 3900 . . . . . . . . . . . 12 (𝑠 ∪ {( 𝐽𝑆)}) = ({( 𝐽𝑆)} ∪ 𝑠)
3836, 37syl6sseq 3792 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑢 ⊆ ({( 𝐽𝑆)} ∪ 𝑠))
39 ssundif 4196 . . . . . . . . . . 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 8357 . . . . . . . . . . . 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 8433 . . . . . . . . . 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 3782 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑠 𝑢)
5046, 49sstrd 3754 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 𝑢)
5150sselda 3744 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) ∧ 𝑣𝑆) → 𝑣 𝑢)
52 eluni 4591 . . . . . . . . . . . . . 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 3877 . . . . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . . . . . . . . . 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 4337 . . . . . . . . . . . . . . . . . . 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 3725 . . . . . . . . . . . . . . . 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 1995 . . . . . . . . . . . . 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 4591 . . . . . . . . . . 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 3750 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) ∧ 𝑠𝐽𝑆 𝑠) ∧ (𝑢 ⊆ (𝑠 ∪ {( 𝐽𝑆)}) ∧ 𝑢 ∈ Fin) ∧ 𝐽 = 𝑢) → 𝑆 (𝑢 ∖ {( 𝐽𝑆)}))
81 unieq 4596 . . . . . . . . . . 11 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → 𝑡 = (𝑢 ∖ {( 𝐽𝑆)}))
8281sseq2d 3774 . . . . . . . . . 10 (𝑡 = (𝑢 ∖ {( 𝐽𝑆)}) → (𝑆 𝑡𝑆 (𝑢 ∖ {( 𝐽𝑆)})))
8382rspcev 3449 . . . . . . . . 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 3171 . . . . . 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 3103 . 2 ((𝐽 ∈ Comp ∧ 𝑆 ∈ (Clsd‘𝐽)) → ∀𝑠 ∈ 𝒫 𝐽(𝑆 𝑠 → ∃𝑡 ∈ (𝒫 𝑠 ∩ Fin)𝑆 𝑡))
91 cmptop 21400 . . 3 (𝐽 ∈ Comp → 𝐽 ∈ Top)
924cldss 21035 . . 3 (𝑆 ∈ (Clsd‘𝐽) → 𝑆 𝐽)
934cmpsub 21405 . . 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 1632  wex 1853  wcel 2139  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cun 3713  cin 3714  wss 3715  𝒫 cpw 4302  {csn 4321   cuni 4588  cfv 6049  (class class class)co 6813  Fincfn 8121  t crest 16283  Topctop 20900  Clsdccld 21022  Compccmp 21391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-en 8122  df-dom 8123  df-fin 8125  df-fi 8482  df-rest 16285  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025  df-cmp 21392
This theorem is referenced by:  hausllycmp  21499  cldllycmp  21500  txkgen  21657  cmphaushmeo  21805  cnheiborlem  22954  cmpcmet  23316  stoweidlem28  40748  stoweidlem50  40770  stoweidlem57  40777
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