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Theorem cmpfi 21405
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 4304 . . . 4 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
2 0ss 4107 . . . . . . . . . . 11 ∅ ⊆ 𝑦
3 0fin 8345 . . . . . . . . . . 11 ∅ ∈ Fin
4 elfpw 8425 . . . . . . . . . . 11 (∅ ∈ (𝒫 𝑦 ∩ Fin) ↔ (∅ ⊆ 𝑦 ∧ ∅ ∈ Fin))
52, 3, 4mpbir2an 993 . . . . . . . . . 10 ∅ ∈ (𝒫 𝑦 ∩ Fin)
6 simprr 813 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝐽 = 𝑦)
7 simprl 811 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝑦 = ∅)
87unieqd 4590 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝑦 = ∅)
96, 8eqtrd 2786 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝐽 = ∅)
10 unieq 4588 . . . . . . . . . . . 12 (𝑧 = ∅ → 𝑧 = ∅)
1110eqeq2d 2762 . . . . . . . . . . 11 (𝑧 = ∅ → ( 𝐽 = 𝑧 𝐽 = ∅))
1211rspcev 3441 . . . . . . . . . 10 ((∅ ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝐽 = ∅) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
135, 9, 12sylancr 698 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
1413expr 644 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
15 vn0 4059 . . . . . . . . . 10 V ≠ ∅
16 iineq1 4679 . . . . . . . . . . . . . 14 (𝑦 = ∅ → 𝑟𝑦 ( 𝐽𝑟) = 𝑟 ∈ ∅ ( 𝐽𝑟))
1716adantl 473 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → 𝑟𝑦 ( 𝐽𝑟) = 𝑟 ∈ ∅ ( 𝐽𝑟))
18 0iin 4722 . . . . . . . . . . . . 13 𝑟 ∈ ∅ ( 𝐽𝑟) = V
1917, 18syl6eq 2802 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → 𝑟𝑦 ( 𝐽𝑟) = V)
2019eqeq1d 2754 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ V = ∅))
2120necon3bbid 2961 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → (¬ 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ V ≠ ∅))
2215, 21mpbiri 248 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ¬ 𝑟𝑦 ( 𝐽𝑟) = ∅)
2322pm2.21d 118 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
2414, 232thd 255 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
25 uniss 4602 . . . . . . . . . . . 12 (𝑦𝐽 𝑦 𝐽)
2625ad2antlr 765 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑦 𝐽)
27 eqss 3751 . . . . . . . . . . . 12 ( 𝑦 = 𝐽 ↔ ( 𝑦 𝐽 𝐽 𝑦))
2827baib 982 . . . . . . . . . . 11 ( 𝑦 𝐽 → ( 𝑦 = 𝐽 𝐽 𝑦))
2926, 28syl 17 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝑦 = 𝐽 𝐽 𝑦))
30 eqcom 2759 . . . . . . . . . 10 ( 𝑦 = 𝐽 𝐽 = 𝑦)
31 ssdif0 4077 . . . . . . . . . 10 ( 𝐽 𝑦 ↔ ( 𝐽 𝑦) = ∅)
3229, 30, 313bitr3g 302 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝐽 = 𝑦 ↔ ( 𝐽 𝑦) = ∅))
33 iindif2 4733 . . . . . . . . . . . 12 (𝑦 ≠ ∅ → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑟𝑦 𝑟))
3433adantl 473 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑟𝑦 𝑟))
35 uniiun 4717 . . . . . . . . . . . 12 𝑦 = 𝑟𝑦 𝑟
3635difeq2i 3860 . . . . . . . . . . 11 ( 𝐽 𝑦) = ( 𝐽 𝑟𝑦 𝑟)
3734, 36syl6eqr 2804 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑦))
3837eqeq1d 2754 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ( 𝐽 𝑦) = ∅))
3932, 38bitr4d 271 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝐽 = 𝑦 𝑟𝑦 ( 𝐽𝑟) = ∅))
40 imassrn 5627 . . . . . . . . . . . 12 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟))
41 df-ima 5271 . . . . . . . . . . . . . 14 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦)
42 resmpt 5599 . . . . . . . . . . . . . . . 16 (𝑦𝐽 → ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = (𝑟𝑦 ↦ ( 𝐽𝑟)))
4342adantl 473 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = (𝑟𝑦 ↦ ( 𝐽𝑟)))
4443rneqd 5500 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ran ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4541, 44syl5eq 2798 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4645ad2antrr 764 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4740, 46syl5sseqr 3787 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
48 funmpt 6079 . . . . . . . . . . . 12 Fun (𝑟𝑦 ↦ ( 𝐽𝑟))
49 elfpw 8425 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑧𝑦𝑧 ∈ Fin))
5049simprbi 483 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 𝑦 ∩ Fin) → 𝑧 ∈ Fin)
5150adantl 473 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → 𝑧 ∈ Fin)
52 imafi 8416 . . . . . . . . . . . 12 ((Fun (𝑟𝑦 ↦ ( 𝐽𝑟)) ∧ 𝑧 ∈ Fin) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin)
5348, 51, 52sylancr 698 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin)
54 elfpw 8425 . . . . . . . . . . 11 (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin) ↔ (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∧ ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin))
5547, 53, 54sylanbrc 701 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin))
56 eqid 2752 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
5756topopn 20905 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → 𝐽𝐽)
58 difexg 4952 . . . . . . . . . . . . . . . 16 ( 𝐽𝐽 → ( 𝐽𝑟) ∈ V)
5957, 58syl 17 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top → ( 𝐽𝑟) ∈ V)
6059ralrimivw 3097 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → ∀𝑟𝑦 ( 𝐽𝑟) ∈ V)
61 eqid 2752 . . . . . . . . . . . . . . 15 (𝑟𝑦 ↦ ( 𝐽𝑟)) = (𝑟𝑦 ↦ ( 𝐽𝑟))
6261fnmpt 6173 . . . . . . . . . . . . . 14 (∀𝑟𝑦 ( 𝐽𝑟) ∈ V → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
6360, 62syl 17 . . . . . . . . . . . . 13 (𝐽 ∈ Top → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
6463ad3antrrr 768 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
65 simpr 479 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin))
66 elfpw 8425 . . . . . . . . . . . . . . 15 (𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin) ↔ (𝑤 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∧ 𝑤 ∈ Fin))
6765, 66sylib 208 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → (𝑤 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∧ 𝑤 ∈ Fin))
6867simpld 477 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
6945ad2antrr 764 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
7068, 69sseqtrd 3774 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
7167simprd 482 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ Fin)
72 fipreima 8429 . . . . . . . . . . . 12 (((𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦𝑤 ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟)) ∧ 𝑤 ∈ Fin) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤)
7364, 70, 71, 72syl3anc 1473 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤)
74 eqcom 2759 . . . . . . . . . . . 12 (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7574rexbii 3171 . . . . . . . . . . 11 (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7673, 75sylib 208 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
77 simpr 479 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7877inteqd 4624 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7978eqeq2d 2762 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → (∅ = 𝑤 ↔ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
8055, 76, 79rexxfrd 5022 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
81 0ex 4934 . . . . . . . . . 10 ∅ ∈ V
82 imassrn 5627 . . . . . . . . . . . . 13 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ⊆ ran (𝑟𝐽 ↦ ( 𝐽𝑟))
83 eqid 2752 . . . . . . . . . . . . . . . . 17 (𝑟𝐽 ↦ ( 𝐽𝑟)) = (𝑟𝐽 ↦ ( 𝐽𝑟))
8456, 83opncldf1 21082 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽1-1-onto→(Clsd‘𝐽) ∧ (𝑟𝐽 ↦ ( 𝐽𝑟)) = (𝑣 ∈ (Clsd‘𝐽) ↦ ( 𝐽𝑣))))
8584simpld 477 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top → (𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽1-1-onto→(Clsd‘𝐽))
86 f1ofo 6297 . . . . . . . . . . . . . . 15 ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽1-1-onto→(Clsd‘𝐽) → (𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽))
8785, 86syl 17 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → (𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽))
88 forn 6271 . . . . . . . . . . . . . 14 ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽) → ran (𝑟𝐽 ↦ ( 𝐽𝑟)) = (Clsd‘𝐽))
8987, 88syl 17 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ran (𝑟𝐽 ↦ ( 𝐽𝑟)) = (Clsd‘𝐽))
9082, 89syl5sseq 3786 . . . . . . . . . . . 12 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
91 fvex 6354 . . . . . . . . . . . . 13 (Clsd‘𝐽) ∈ V
9291elpw2 4969 . . . . . . . . . . . 12 (((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽) ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
9390, 92sylibr 224 . . . . . . . . . . 11 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
9493ad2antrr 764 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
95 elfi 8476 . . . . . . . . . 10 ((∅ ∈ V ∧ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽)) → (∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤))
9681, 94, 95sylancr 698 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤))
97 inundif 4182 . . . . . . . . . . . . . 14 (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) = (𝒫 𝑦 ∩ Fin)
9897rexeqi 3274 . . . . . . . . . . . . 13 (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
99 rexun 3928 . . . . . . . . . . . . 13 (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) 𝐽 = 𝑧 ↔ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
10098, 99bitr3i 266 . . . . . . . . . . . 12 (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
101 inss2 3969 . . . . . . . . . . . . . . . . . . . . . 22 ((𝒫 𝑦 ∩ Fin) ∩ {∅}) ⊆ {∅}
102101sseli 3732 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 ∈ {∅})
103 elsni 4330 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ {∅} → 𝑧 = ∅)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
105104unieqd 4590 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
106 uni0 4609 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
107105, 106syl6eq 2802 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
108107eqeq2d 2762 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ( 𝐽 = 𝑧 𝐽 = ∅))
109108biimpd 219 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ( 𝐽 = 𝑧 𝐽 = ∅))
110109rexlimiv 3157 . . . . . . . . . . . . . . 15 (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 𝐽 = ∅)
111 ssid 3757 . . . . . . . . . . . . . . . . . . . 20 𝑦𝑦
112111a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦𝑦)
113 simprr 813 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 = ∅)
114 0ss 4107 . . . . . . . . . . . . . . . . . . . . . . . . 25 ∅ ⊆ 𝑦
115113, 114syl6eqss 3788 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 𝑦)
11625ad2antlr 765 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 𝐽)
117115, 116eqssd 3753 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 = 𝑦)
118117, 113eqtr3d 2788 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 = ∅)
119118, 3syl6eqel 2839 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ Fin)
120 pwfi 8418 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
121119, 120sylib 208 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝒫 𝑦 ∈ Fin)
122 pwuni 4618 . . . . . . . . . . . . . . . . . . . 20 𝑦 ⊆ 𝒫 𝑦
123 ssfi 8337 . . . . . . . . . . . . . . . . . . . 20 ((𝒫 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝒫 𝑦) → 𝑦 ∈ Fin)
124121, 122, 123sylancl 697 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ Fin)
125 elfpw 8425 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦𝑦𝑦 ∈ Fin))
126112, 124, 125sylanbrc 701 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ (𝒫 𝑦 ∩ Fin))
127 simprl 811 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ≠ ∅)
128 eldifsn 4454 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑦 ≠ ∅))
129126, 127, 128sylanbrc 701 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))
130 unieq 4588 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑦 𝑧 = 𝑦)
131130eqeq2d 2762 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → ( 𝐽 = 𝑧 𝐽 = 𝑦))
132131rspcev 3441 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ 𝐽 = 𝑦) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧)
133129, 117, 132syl2anc 696 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧)
134133expr 644 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝐽 = ∅ → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
135110, 134syl5 34 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
136 idd 24 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧 → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
137135, 136jaod 394 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ((∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
138 olc 398 . . . . . . . . . . . . 13 (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧 → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
139137, 138impbid1 215 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ((∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 ∨ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧) ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
140100, 139syl5bb 272 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧))
141 eldifi 3867 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
142141adantl 473 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
143142, 49sylib 208 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (𝑧𝑦𝑧 ∈ Fin))
144143simpld 477 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧𝑦)
145 simpllr 817 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑦𝐽)
146144, 145sstrd 3746 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧𝐽)
147146unissd 4606 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 𝐽)
148 eqss 3751 . . . . . . . . . . . . . . . 16 ( 𝑧 = 𝐽 ↔ ( 𝑧 𝐽 𝐽 𝑧))
149148baib 982 . . . . . . . . . . . . . . 15 ( 𝑧 𝐽 → ( 𝑧 = 𝐽 𝐽 𝑧))
150147, 149syl 17 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ( 𝑧 = 𝐽 𝐽 𝑧))
151 eqcom 2759 . . . . . . . . . . . . . 14 ( 𝑧 = 𝐽 𝐽 = 𝑧)
152 ssdif0 4077 . . . . . . . . . . . . . . 15 ( 𝐽 𝑧 ↔ ( 𝐽 𝑧) = ∅)
153 eqcom 2759 . . . . . . . . . . . . . . 15 (( 𝐽 𝑧) = ∅ ↔ ∅ = ( 𝐽 𝑧))
154152, 153bitri 264 . . . . . . . . . . . . . 14 ( 𝐽 𝑧 ↔ ∅ = ( 𝐽 𝑧))
155150, 151, 1543bitr3g 302 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ( 𝐽 = 𝑧 ↔ ∅ = ( 𝐽 𝑧)))
156 df-ima 5271 . . . . . . . . . . . . . . . . . 18 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧)
157144resmptd 5602 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧) = (𝑟𝑧 ↦ ( 𝐽𝑟)))
158157rneqd 5500 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ran ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
159156, 158syl5eq 2798 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
160159inteqd 4624 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
16159ralrimivw 3097 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ Top → ∀𝑟𝑧 ( 𝐽𝑟) ∈ V)
162161ad3antrrr 768 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∀𝑟𝑧 ( 𝐽𝑟) ∈ V)
163 dfiin3g 5526 . . . . . . . . . . . . . . . . 17 (∀𝑟𝑧 ( 𝐽𝑟) ∈ V → 𝑟𝑧 ( 𝐽𝑟) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
164162, 163syl 17 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑟𝑧 ( 𝐽𝑟) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
165 eldifsn 4454 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑧 ≠ ∅))
166165simprbi 483 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
167166adantl 473 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ≠ ∅)
168 iindif2 4733 . . . . . . . . . . . . . . . . 17 (𝑧 ≠ ∅ → 𝑟𝑧 ( 𝐽𝑟) = ( 𝐽 𝑟𝑧 𝑟))
169167, 168syl 17 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑟𝑧 ( 𝐽𝑟) = ( 𝐽 𝑟𝑧 𝑟))
170160, 164, 1693eqtr2d 2792 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ( 𝐽 𝑟𝑧 𝑟))
171 uniiun 4717 . . . . . . . . . . . . . . . 16 𝑧 = 𝑟𝑧 𝑟
172171difeq2i 3860 . . . . . . . . . . . . . . 15 ( 𝐽 𝑧) = ( 𝐽 𝑟𝑧 𝑟)
173170, 172syl6eqr 2804 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ( 𝐽 𝑧))
174173eqeq2d 2762 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → (∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ↔ ∅ = ( 𝐽 𝑧)))
175155, 174bitr4d 271 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ( 𝐽 = 𝑧 ↔ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
176175rexbidva 3179 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
177140, 176bitrd 268 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
178 imaeq2 5612 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ ∅))
179 ima0 5631 . . . . . . . . . . . . . . . . . . . 20 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ ∅) = ∅
180178, 179syl6eq 2802 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ∅)
181180inteqd 4624 . . . . . . . . . . . . . . . . . 18 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ∅)
182 int0 4634 . . . . . . . . . . . . . . . . . 18 ∅ = V
183181, 182syl6eq 2802 . . . . . . . . . . . . . . . . 17 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = V)
184183neeq1d 2983 . . . . . . . . . . . . . . . 16 (𝑧 = ∅ → ( ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ≠ ∅ ↔ V ≠ ∅))
18515, 184mpbiri 248 . . . . . . . . . . . . . . 15 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ≠ ∅)
186185necomd 2979 . . . . . . . . . . . . . 14 (𝑧 = ∅ → ∅ ≠ ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
187186necon2i 2958 . . . . . . . . . . . . 13 (∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) → 𝑧 ≠ ∅)
188165rbaibr 984 . . . . . . . . . . . . 13 (𝑧 ≠ ∅ → (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})))
189187, 188syl 17 . . . . . . . . . . . 12 (∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) → (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})))
190189pm5.32ri 673 . . . . . . . . . . 11 ((𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) ↔ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
191190rexbii2 3169 . . . . . . . . . 10 (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ↔ ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
192177, 191syl6bbr 278 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
19380, 96, 1923bitr4rd 301 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧 ↔ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
19439, 193imbi12d 333 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
19524, 194pm2.61dane 3011 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
19660adantr 472 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ∀𝑟𝑦 ( 𝐽𝑟) ∈ V)
197 dfiin3g 5526 . . . . . . . . . . 11 (∀𝑟𝑦 ( 𝐽𝑟) ∈ V → 𝑟𝑦 ( 𝐽𝑟) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
198196, 197syl 17 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑟𝑦 ( 𝐽𝑟) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
19945inteqd 4624 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
200198, 199eqtr4d 2789 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑟𝑦 ( 𝐽𝑟) = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
201200eqeq1d 2754 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ∅))
202 nne 2928 . . . . . . . 8 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ∅)
203201, 202syl6bbr 278 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅))
204203imbi1d 330 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))) ↔ (¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
205195, 204bitrd 268 . . . . 5 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ (¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
206 con1b 347 . . . . 5 ((¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅))
207205, 206syl6bb 276 . . . 4 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
2081, 207sylan2 492 . . 3 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
209208ralbidva 3115 . 2 (𝐽 ∈ Top → (∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ∀𝑦 ∈ 𝒫 𝐽(¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
21056iscmp 21385 . . 3 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
211210baib 982 . 2 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
21293adantr 472 . . 3 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
213 simpl 474 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝐽 ∈ Top)
214 funmpt 6079 . . . . . 6 Fun (𝑟𝐽 ↦ ( 𝐽𝑟))
215214a1i 11 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → Fun (𝑟𝐽 ↦ ( 𝐽𝑟)))
216 elpwi 4304 . . . . . . 7 (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
217 foima 6273 . . . . . . . . 9 ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽) = (Clsd‘𝐽))
21887, 217syl 17 . . . . . . . 8 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽) = (Clsd‘𝐽))
219218sseq2d 3766 . . . . . . 7 (𝐽 ∈ Top → (𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽) ↔ 𝑥 ⊆ (Clsd‘𝐽)))
220216, 219syl5ibr 236 . . . . . 6 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽)))
221220imp 444 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽))
222 ssimaexg 6418 . . . . 5 ((𝐽 ∈ Top ∧ Fun (𝑟𝐽 ↦ ( 𝐽𝑟)) ∧ 𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽)) → ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
223213, 215, 221, 222syl3anc 1473 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
224 df-rex 3048 . . . . 5 (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
225 selpw 4301 . . . . . . 7 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
226225anbi1i 733 . . . . . 6 ((𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ (𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
227226exbii 1915 . . . . 5 (∃𝑦(𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
228224, 227bitri 264 . . . 4 (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
229223, 228sylibr 224 . . 3 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
230 simpr 479 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
231230fveq2d 6348 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (fi‘𝑥) = (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
232231eleq2d 2817 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (∅ ∈ (fi‘𝑥) ↔ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
233232notbid 307 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (¬ ∅ ∈ (fi‘𝑥) ↔ ¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
234230inteqd 4624 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
235234neeq1d 2983 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ( 𝑥 ≠ ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅))
236233, 235imbi12d 333 . . 3 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
237212, 229, 236ralxfrd 5020 . 2 (𝐽 ∈ Top → (∀𝑥 ∈ 𝒫 (Clsd‘𝐽)(¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅) ↔ ∀𝑦 ∈ 𝒫 𝐽(¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
238209, 211, 2373bitr4d 300 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 196  wo 382  wa 383   = wceq 1624  wex 1845  wcel 2131  wne 2924  wral 3042  wrex 3043  Vcvv 3332  cdif 3704  cun 3705  cin 3706  wss 3707  c0 4050  𝒫 cpw 4294  {csn 4313   cuni 4580   cint 4619   ciun 4664   ciin 4665  cmpt 4873  ccnv 5257  ran crn 5259  cres 5260  cima 5261  Fun wfun 6035   Fn wfn 6036  ontowfo 6039  1-1-ontowf1o 6040  cfv 6041  Fincfn 8113  ficfi 8473  Topctop 20892  Clsdccld 21014  Compccmp 21383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-iin 4667  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-1st 7325  df-2nd 7326  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-1o 7721  df-2o 7722  df-oadd 7725  df-er 7903  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-fi 8474  df-top 20893  df-cld 21017  df-cmp 21384
This theorem is referenced by:  cmpfii  21406  fclscmp  22027  heibor1lem  33913
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