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Theorem cmpfi 22642
Description: If a topology is compact and a collection of closed sets has the finite intersection property, its intersection is nonempty. (Contributed by Jeff Hankins, 25-Aug-2009.) (Proof shortened by Mario Carneiro, 1-Sep-2015.)
Assertion
Ref Expression
cmpfi (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
Distinct variable group:   𝑥,𝐽

Proof of Theorem cmpfi
Dummy variables 𝑣 𝑟 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 4552 . . . 4 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
2 0ss 4341 . . . . . . . . . . 11 ∅ ⊆ 𝑦
3 0fin 9015 . . . . . . . . . . 11 ∅ ∈ Fin
4 elfpw 9198 . . . . . . . . . . 11 (∅ ∈ (𝒫 𝑦 ∩ Fin) ↔ (∅ ⊆ 𝑦 ∧ ∅ ∈ Fin))
52, 3, 4mpbir2an 708 . . . . . . . . . 10 ∅ ∈ (𝒫 𝑦 ∩ Fin)
6 simprr 770 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝐽 = 𝑦)
7 simprl 768 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝑦 = ∅)
87unieqd 4864 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝑦 = ∅)
96, 8eqtrd 2777 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝐽 = ∅)
10 unieq 4861 . . . . . . . . . . 11 (𝑧 = ∅ → 𝑧 = ∅)
1110rspceeqv 3584 . . . . . . . . . 10 ((∅ ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝐽 = ∅) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
125, 9, 11sylancr 587 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
1312expr 457 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
14 vn0 4283 . . . . . . . . . 10 V ≠ ∅
15 iineq1 4954 . . . . . . . . . . . . . 14 (𝑦 = ∅ → 𝑟𝑦 ( 𝐽𝑟) = 𝑟 ∈ ∅ ( 𝐽𝑟))
1615adantl 482 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → 𝑟𝑦 ( 𝐽𝑟) = 𝑟 ∈ ∅ ( 𝐽𝑟))
17 0iin 5006 . . . . . . . . . . . . 13 𝑟 ∈ ∅ ( 𝐽𝑟) = V
1816, 17eqtrdi 2793 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → 𝑟𝑦 ( 𝐽𝑟) = V)
1918eqeq1d 2739 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ V = ∅))
2019necon3bbid 2979 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → (¬ 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ V ≠ ∅))
2114, 20mpbiri 257 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ¬ 𝑟𝑦 ( 𝐽𝑟) = ∅)
2221pm2.21d 121 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
2313, 222thd 264 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
24 uniss 4858 . . . . . . . . . . . 12 (𝑦𝐽 𝑦 𝐽)
2524ad2antlr 724 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑦 𝐽)
26 eqss 3946 . . . . . . . . . . . 12 ( 𝑦 = 𝐽 ↔ ( 𝑦 𝐽 𝐽 𝑦))
2726baib 536 . . . . . . . . . . 11 ( 𝑦 𝐽 → ( 𝑦 = 𝐽 𝐽 𝑦))
2825, 27syl 17 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝑦 = 𝐽 𝐽 𝑦))
29 eqcom 2744 . . . . . . . . . 10 ( 𝑦 = 𝐽 𝐽 = 𝑦)
30 ssdif0 4308 . . . . . . . . . 10 ( 𝐽 𝑦 ↔ ( 𝐽 𝑦) = ∅)
3128, 29, 303bitr3g 312 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝐽 = 𝑦 ↔ ( 𝐽 𝑦) = ∅))
32 iindif2 5019 . . . . . . . . . . . 12 (𝑦 ≠ ∅ → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑟𝑦 𝑟))
3332adantl 482 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑟𝑦 𝑟))
34 uniiun 5001 . . . . . . . . . . . 12 𝑦 = 𝑟𝑦 𝑟
3534difeq2i 4065 . . . . . . . . . . 11 ( 𝐽 𝑦) = ( 𝐽 𝑟𝑦 𝑟)
3633, 35eqtr4di 2795 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑦))
3736eqeq1d 2739 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ( 𝐽 𝑦) = ∅))
3831, 37bitr4d 281 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝐽 = 𝑦 𝑟𝑦 ( 𝐽𝑟) = ∅))
39 imassrn 5998 . . . . . . . . . . . 12 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟))
40 df-ima 5621 . . . . . . . . . . . . . 14 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦)
41 resmpt 5965 . . . . . . . . . . . . . . . 16 (𝑦𝐽 → ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = (𝑟𝑦 ↦ ( 𝐽𝑟)))
4241adantl 482 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = (𝑟𝑦 ↦ ( 𝐽𝑟)))
4342rneqd 5867 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ran ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4440, 43eqtrid 2789 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4544ad2antrr 723 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4639, 45sseqtrrid 3984 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
47 funmpt 6509 . . . . . . . . . . . 12 Fun (𝑟𝑦 ↦ ( 𝐽𝑟))
48 elinel2 4141 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 𝑦 ∩ Fin) → 𝑧 ∈ Fin)
4948adantl 482 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → 𝑧 ∈ Fin)
50 imafi 9019 . . . . . . . . . . . 12 ((Fun (𝑟𝑦 ↦ ( 𝐽𝑟)) ∧ 𝑧 ∈ Fin) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin)
5147, 49, 50sylancr 587 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin)
52 elfpw 9198 . . . . . . . . . . 11 (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin) ↔ (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∧ ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin))
5346, 51, 52sylanbrc 583 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin))
54 eqid 2737 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
5554topopn 22138 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → 𝐽𝐽)
5655difexd 5268 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top → ( 𝐽𝑟) ∈ V)
5756ralrimivw 3144 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → ∀𝑟𝑦 ( 𝐽𝑟) ∈ V)
58 eqid 2737 . . . . . . . . . . . . . . 15 (𝑟𝑦 ↦ ( 𝐽𝑟)) = (𝑟𝑦 ↦ ( 𝐽𝑟))
5958fnmpt 6611 . . . . . . . . . . . . . 14 (∀𝑟𝑦 ( 𝐽𝑟) ∈ V → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
6057, 59syl 17 . . . . . . . . . . . . 13 (𝐽 ∈ Top → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
6160ad3antrrr 727 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
62 simpr 485 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin))
63 elfpw 9198 . . . . . . . . . . . . . . 15 (𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin) ↔ (𝑤 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∧ 𝑤 ∈ Fin))
6462, 63sylib 217 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → (𝑤 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∧ 𝑤 ∈ Fin))
6564simpld 495 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
6644ad2antrr 723 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
6765, 66sseqtrd 3971 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
6864simprd 496 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ Fin)
69 fipreima 9202 . . . . . . . . . . . 12 (((𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦𝑤 ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟)) ∧ 𝑤 ∈ Fin) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤)
7061, 67, 68, 69syl3anc 1370 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤)
71 eqcom 2744 . . . . . . . . . . . 12 (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7271rexbii 3094 . . . . . . . . . . 11 (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7370, 72sylib 217 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
74 simpr 485 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7574inteqd 4897 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7675eqeq2d 2748 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → (∅ = 𝑤 ↔ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
7753, 73, 76rexxfrd 5347 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
78 0ex 5246 . . . . . . . . . 10 ∅ ∈ V
79 imassrn 5998 . . . . . . . . . . . . 13 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ⊆ ran (𝑟𝐽 ↦ ( 𝐽𝑟))
80 eqid 2737 . . . . . . . . . . . . . . . . 17 (𝑟𝐽 ↦ ( 𝐽𝑟)) = (𝑟𝐽 ↦ ( 𝐽𝑟))
8154, 80opncldf1 22318 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽1-1-onto→(Clsd‘𝐽) ∧ (𝑟𝐽 ↦ ( 𝐽𝑟)) = (𝑣 ∈ (Clsd‘𝐽) ↦ ( 𝐽𝑣))))
8281simpld 495 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top → (𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽1-1-onto→(Clsd‘𝐽))
83 f1ofo 6761 . . . . . . . . . . . . . . 15 ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽1-1-onto→(Clsd‘𝐽) → (𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽))
8482, 83syl 17 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → (𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽))
85 forn 6729 . . . . . . . . . . . . . 14 ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽) → ran (𝑟𝐽 ↦ ( 𝐽𝑟)) = (Clsd‘𝐽))
8684, 85syl 17 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ran (𝑟𝐽 ↦ ( 𝐽𝑟)) = (Clsd‘𝐽))
8779, 86sseqtrid 3983 . . . . . . . . . . . 12 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
88 fvex 6825 . . . . . . . . . . . . 13 (Clsd‘𝐽) ∈ V
8988elpw2 5284 . . . . . . . . . . . 12 (((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽) ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
9087, 89sylibr 233 . . . . . . . . . . 11 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
9190ad2antrr 723 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
92 elfi 9249 . . . . . . . . . 10 ((∅ ∈ V ∧ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽)) → (∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤))
9378, 91, 92sylancr 587 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤))
94 inundif 4423 . . . . . . . . . . . . . 14 (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) = (𝒫 𝑦 ∩ Fin)
9594rexeqi 3309 . . . . . . . . . . . . 13 (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
96 rexun 4135 . . . . . . . . . . . . 13 (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) 𝐽 = 𝑧 ↔ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
9795, 96bitr3i 276 . . . . . . . . . . . 12 (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
98 elinel2 4141 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 ∈ {∅})
99 elsni 4588 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ {∅} → 𝑧 = ∅)
10098, 99syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
101100unieqd 4864 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
102 uni0 4881 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
103101, 102eqtrdi 2793 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
104103eqeq2d 2748 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ( 𝐽 = 𝑧 𝐽 = ∅))
105104biimpd 228 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ( 𝐽 = 𝑧 𝐽 = ∅))
106105rexlimiv 3142 . . . . . . . . . . . . . . 15 (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 𝐽 = ∅)
107 ssidd 3954 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦𝑦)
108 simprr 770 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 = ∅)
109 0ss 4341 . . . . . . . . . . . . . . . . . . . . . . . . 25 ∅ ⊆ 𝑦
110108, 109eqsstrdi 3985 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 𝑦)
11124ad2antlr 724 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 𝐽)
112110, 111eqssd 3948 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 = 𝑦)
113112, 108eqtr3d 2779 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 = ∅)
114113, 3eqeltrdi 2846 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ Fin)
115 pwfi 9022 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
116114, 115sylib 217 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝒫 𝑦 ∈ Fin)
117 pwuni 4891 . . . . . . . . . . . . . . . . . . . 20 𝑦 ⊆ 𝒫 𝑦
118 ssfi 9017 . . . . . . . . . . . . . . . . . . . 20 ((𝒫 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝒫 𝑦) → 𝑦 ∈ Fin)
119116, 117, 118sylancl 586 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ Fin)
120 elfpw 9198 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦𝑦𝑦 ∈ Fin))
121107, 119, 120sylanbrc 583 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ (𝒫 𝑦 ∩ Fin))
122 simprl 768 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ≠ ∅)
123 eldifsn 4732 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑦 ≠ ∅))
124121, 122, 123sylanbrc 583 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))
125 unieq 4861 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 𝑧 = 𝑦)
126125rspceeqv 3584 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ 𝐽 = 𝑦) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧)
127124, 112, 126syl2anc 584 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧)
128127expr 457 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝐽 = ∅ → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
129106, 128syl5 34 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
130 idd 24 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧 → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
131129, 130jaod 856 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ((∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
132 olc 865 . . . . . . . . . . . . 13 (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧 → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
133131, 132impbid1 224 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ((∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧) ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
13497, 133bitrid 282 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
135 eldifi 4072 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
136135adantl 482 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
137 elfpw 9198 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑧𝑦𝑧 ∈ Fin))
138136, 137sylib 217 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (𝑧𝑦𝑧 ∈ Fin))
139138simpld 495 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧𝑦)
140 simpllr 773 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑦𝐽)
141139, 140sstrd 3941 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧𝐽)
142141unissd 4860 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 𝐽)
143 eqss 3946 . . . . . . . . . . . . . . . 16 ( 𝑧 = 𝐽 ↔ ( 𝑧 𝐽 𝐽 𝑧))
144143baib 536 . . . . . . . . . . . . . . 15 ( 𝑧 𝐽 → ( 𝑧 = 𝐽 𝐽 𝑧))
145142, 144syl 17 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ( 𝑧 = 𝐽 𝐽 𝑧))
146 eqcom 2744 . . . . . . . . . . . . . 14 ( 𝑧 = 𝐽 𝐽 = 𝑧)
147 ssdif0 4308 . . . . . . . . . . . . . . 15 ( 𝐽 𝑧 ↔ ( 𝐽 𝑧) = ∅)
148 eqcom 2744 . . . . . . . . . . . . . . 15 (( 𝐽 𝑧) = ∅ ↔ ∅ = ( 𝐽 𝑧))
149147, 148bitri 274 . . . . . . . . . . . . . 14 ( 𝐽 𝑧 ↔ ∅ = ( 𝐽 𝑧))
150145, 146, 1493bitr3g 312 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ( 𝐽 = 𝑧 ↔ ∅ = ( 𝐽 𝑧)))
151 df-ima 5621 . . . . . . . . . . . . . . . . . 18 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧)
152139resmptd 5968 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧) = (𝑟𝑧 ↦ ( 𝐽𝑟)))
153152rneqd 5867 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ran ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
154151, 153eqtrid 2789 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
155154inteqd 4897 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
15656ralrimivw 3144 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ Top → ∀𝑟𝑧 ( 𝐽𝑟) ∈ V)
157156ad3antrrr 727 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∀𝑟𝑧 ( 𝐽𝑟) ∈ V)
158 dfiin3g 5894 . . . . . . . . . . . . . . . . 17 (∀𝑟𝑧 ( 𝐽𝑟) ∈ V → 𝑟𝑧 ( 𝐽𝑟) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
159157, 158syl 17 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑟𝑧 ( 𝐽𝑟) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
160 eldifsni 4735 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
161160adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ≠ ∅)
162 iindif2 5019 . . . . . . . . . . . . . . . . 17 (𝑧 ≠ ∅ → 𝑟𝑧 ( 𝐽𝑟) = ( 𝐽 𝑟𝑧 𝑟))
163161, 162syl 17 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑟𝑧 ( 𝐽𝑟) = ( 𝐽 𝑟𝑧 𝑟))
164155, 159, 1633eqtr2d 2783 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ( 𝐽 𝑟𝑧 𝑟))
165 uniiun 5001 . . . . . . . . . . . . . . . 16 𝑧 = 𝑟𝑧 𝑟
166165difeq2i 4065 . . . . . . . . . . . . . . 15 ( 𝐽 𝑧) = ( 𝐽 𝑟𝑧 𝑟)
167164, 166eqtr4di 2795 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ( 𝐽 𝑧))
168167eqeq2d 2748 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ↔ ∅ = ( 𝐽 𝑧)))
169150, 168bitr4d 281 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ( 𝐽 = 𝑧 ↔ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
170169rexbidva 3170 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
171134, 170bitrd 278 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
172 imaeq2 5983 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ ∅))
173 ima0 6003 . . . . . . . . . . . . . . . . . . . 20 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ ∅) = ∅
174172, 173eqtrdi 2793 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ∅)
175174inteqd 4897 . . . . . . . . . . . . . . . . . 18 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ∅)
176 int0 4906 . . . . . . . . . . . . . . . . . 18 ∅ = V
177175, 176eqtrdi 2793 . . . . . . . . . . . . . . . . 17 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = V)
178177neeq1d 3001 . . . . . . . . . . . . . . . 16 (𝑧 = ∅ → ( ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ≠ ∅ ↔ V ≠ ∅))
17914, 178mpbiri 257 . . . . . . . . . . . . . . 15 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ≠ ∅)
180179necomd 2997 . . . . . . . . . . . . . 14 (𝑧 = ∅ → ∅ ≠ ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
181180necon2i 2976 . . . . . . . . . . . . 13 (∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) → 𝑧 ≠ ∅)
182 eldifsn 4732 . . . . . . . . . . . . . 14 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑧 ≠ ∅))
183182rbaibr 538 . . . . . . . . . . . . 13 (𝑧 ≠ ∅ → (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})))
184181, 183syl 17 . . . . . . . . . . . 12 (∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) → (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})))
185184pm5.32ri 576 . . . . . . . . . . 11 ((𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) ↔ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
186185rexbii2 3090 . . . . . . . . . 10 (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
187171, 186bitr4di 288 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
18877, 93, 1873bitr4rd 311 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
18938, 188imbi12d 344 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
19023, 189pm2.61dane 3030 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
19157adantr 481 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ∀𝑟𝑦 ( 𝐽𝑟) ∈ V)
192 dfiin3g 5894 . . . . . . . . . . 11 (∀𝑟𝑦 ( 𝐽𝑟) ∈ V → 𝑟𝑦 ( 𝐽𝑟) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
193191, 192syl 17 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑟𝑦 ( 𝐽𝑟) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
19444inteqd 4897 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
195193, 194eqtr4d 2780 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑟𝑦 ( 𝐽𝑟) = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
196195eqeq1d 2739 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ∅))
197 nne 2945 . . . . . . . 8 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ∅)
198196, 197bitr4di 288 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅))
199198imbi1d 341 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))) ↔ (¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
200190, 199bitrd 278 . . . . 5 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ (¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
201 con1b 358 . . . . 5 ((¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅))
202200, 201bitrdi 286 . . . 4 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
2031, 202sylan2 593 . . 3 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
204203ralbidva 3169 . 2 (𝐽 ∈ Top → (∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝐽(¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
20554iscmp 22622 . . 3 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
206205baib 536 . 2 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
20790adantr 481 . . 3 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
208 simpl 483 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝐽 ∈ Top)
209 funmpt 6509 . . . . . 6 Fun (𝑟𝐽 ↦ ( 𝐽𝑟))
210209a1i 11 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → Fun (𝑟𝐽 ↦ ( 𝐽𝑟)))
211 elpwi 4552 . . . . . . 7 (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
212 foima 6731 . . . . . . . . 9 ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽) = (Clsd‘𝐽))
21384, 212syl 17 . . . . . . . 8 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽) = (Clsd‘𝐽))
214213sseq2d 3963 . . . . . . 7 (𝐽 ∈ Top → (𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽) ↔ 𝑥 ⊆ (Clsd‘𝐽)))
215211, 214syl5ibr 245 . . . . . 6 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽)))
216215imp 407 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽))
217 ssimaexg 6894 . . . . 5 ((𝐽 ∈ Top ∧ Fun (𝑟𝐽 ↦ ( 𝐽𝑟)) ∧ 𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽)) → ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
218208, 210, 216, 217syl3anc 1370 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
219 df-rex 3072 . . . . 5 (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
220 velpw 4550 . . . . . . 7 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
221220anbi1i 624 . . . . . 6 ((𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ (𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
222221exbii 1849 . . . . 5 (∃𝑦(𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
223219, 222bitri 274 . . . 4 (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
224218, 223sylibr 233 . . 3 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
225 simpr 485 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
226225fveq2d 6816 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (fi‘𝑥) = (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
227226eleq2d 2823 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (∅ ∈ (fi‘𝑥) ↔ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
228227notbid 317 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (¬ ∅ ∈ (fi‘𝑥) ↔ ¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
229225inteqd 4897 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
230229neeq1d 3001 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ( 𝑥 ≠ ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅))
231228, 230imbi12d 344 . . 3 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
232207, 224, 231ralxfrd 5346 . 2 (𝐽 ∈ Top → (∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅) ↔ ∀𝑦 ∈ 𝒫 𝐽(¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
233204, 206, 2323bitr4d 310 1 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1540  wex 1780  wcel 2105  wne 2941  wral 3062  wrex 3071  Vcvv 3441  cdif 3894  cun 3895  cin 3896  wss 3897  c0 4267  𝒫 cpw 4545  {csn 4571   cuni 4850   cint 4892   ciun 4937   ciin 4938  cmpt 5170  ccnv 5607  ran crn 5609  cres 5610  cima 5611  Fun wfun 6460   Fn wfn 6461  ontowfo 6464  1-1-ontowf1o 6465  cfv 6466  Fincfn 8783  ficfi 9246  Topctop 22125  Clsdccld 22250  Compccmp 22620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-int 4893  df-iun 4939  df-iin 4940  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-om 7760  df-1o 8346  df-en 8784  df-fin 8787  df-fi 9247  df-top 22126  df-cld 22253  df-cmp 22621
This theorem is referenced by:  cmpfii  22643  fclscmp  23264  zarcmplem  31971  heibor1lem  36039
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