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Theorem cmpfi 21121
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 4140 . . . 4 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
2 0ss 3944 . . . . . . . . . . 11 ∅ ⊆ 𝑦
3 0fin 8132 . . . . . . . . . . 11 ∅ ∈ Fin
4 elfpw 8212 . . . . . . . . . . 11 (∅ ∈ (𝒫 𝑦 ∩ Fin) ↔ (∅ ⊆ 𝑦 ∧ ∅ ∈ Fin))
52, 3, 4mpbir2an 954 . . . . . . . . . 10 ∅ ∈ (𝒫 𝑦 ∩ Fin)
6 simprr 795 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝐽 = 𝑦)
7 simprl 793 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝑦 = ∅)
87unieqd 4412 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝑦 = ∅)
96, 8eqtrd 2655 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → 𝐽 = ∅)
10 unieq 4410 . . . . . . . . . . . 12 (𝑧 = ∅ → 𝑧 = ∅)
1110eqeq2d 2631 . . . . . . . . . . 11 (𝑧 = ∅ → ( 𝐽 = 𝑧 𝐽 = ∅))
1211rspcev 3295 . . . . . . . . . 10 ((∅ ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝐽 = ∅) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
135, 9, 12sylancr 694 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 = ∅ ∧ 𝐽 = 𝑦)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
1413expr 642 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧))
15 vn0 3900 . . . . . . . . . 10 V ≠ ∅
16 iineq1 4501 . . . . . . . . . . . . . 14 (𝑦 = ∅ → 𝑟𝑦 ( 𝐽𝑟) = 𝑟 ∈ ∅ ( 𝐽𝑟))
1716adantl 482 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → 𝑟𝑦 ( 𝐽𝑟) = 𝑟 ∈ ∅ ( 𝐽𝑟))
18 0iin 4544 . . . . . . . . . . . . 13 𝑟 ∈ ∅ ( 𝐽𝑟) = V
1917, 18syl6eq 2671 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → 𝑟𝑦 ( 𝐽𝑟) = V)
2019eqeq1d 2623 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 = ∅) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ V = ∅))
2120necon3bbid 2827 . . . . . . . . . 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 4424 . . . . . . . . . . . 12 (𝑦𝐽 𝑦 𝐽)
2625ad2antlr 762 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑦 𝐽)
27 eqss 3598 . . . . . . . . . . . 12 ( 𝑦 = 𝐽 ↔ ( 𝑦 𝐽 𝐽 𝑦))
2827baib 943 . . . . . . . . . . 11 ( 𝑦 𝐽 → ( 𝑦 = 𝐽 𝐽 𝑦))
2926, 28syl 17 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝑦 = 𝐽 𝐽 𝑦))
30 eqcom 2628 . . . . . . . . . 10 ( 𝑦 = 𝐽 𝐽 = 𝑦)
31 ssdif0 3916 . . . . . . . . . 10 ( 𝐽 𝑦 ↔ ( 𝐽 𝑦) = ∅)
3229, 30, 313bitr3g 302 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝐽 = 𝑦 ↔ ( 𝐽 𝑦) = ∅))
33 iindif2 4555 . . . . . . . . . . . 12 (𝑦 ≠ ∅ → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑟𝑦 𝑟))
3433adantl 482 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑟𝑦 𝑟))
35 uniiun 4539 . . . . . . . . . . . 12 𝑦 = 𝑟𝑦 𝑟
3635difeq2i 3703 . . . . . . . . . . 11 ( 𝐽 𝑦) = ( 𝐽 𝑟𝑦 𝑟)
3734, 36syl6eqr 2673 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → 𝑟𝑦 ( 𝐽𝑟) = ( 𝐽 𝑦))
3837eqeq1d 2623 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ( 𝐽 𝑦) = ∅))
3932, 38bitr4d 271 . . . . . . . 8 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ( 𝐽 = 𝑦 𝑟𝑦 ( 𝐽𝑟) = ∅))
40 imassrn 5436 . . . . . . . . . . . 12 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟))
41 df-ima 5087 . . . . . . . . . . . . . 14 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦)
42 resmpt 5408 . . . . . . . . . . . . . . . 16 (𝑦𝐽 → ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = (𝑟𝑦 ↦ ( 𝐽𝑟)))
4342adantl 482 . . . . . . . . . . . . . . 15 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = (𝑟𝑦 ↦ ( 𝐽𝑟)))
4443rneqd 5313 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ran ((𝑟𝐽 ↦ ( 𝐽𝑟)) ↾ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4541, 44syl5eq 2667 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4645ad2antrr 761 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
4740, 46syl5sseqr 3633 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
48 funmpt 5884 . . . . . . . . . . . 12 Fun (𝑟𝑦 ↦ ( 𝐽𝑟))
49 elfpw 8212 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑧𝑦𝑧 ∈ Fin))
5049simprbi 480 . . . . . . . . . . . . 13 (𝑧 ∈ (𝒫 𝑦 ∩ Fin) → 𝑧 ∈ Fin)
5150adantl 482 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → 𝑧 ∈ Fin)
52 imafi 8203 . . . . . . . . . . . 12 ((Fun (𝑟𝑦 ↦ ( 𝐽𝑟)) ∧ 𝑧 ∈ Fin) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin)
5348, 51, 52sylancr 694 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin)
54 elfpw 8212 . . . . . . . . . . 11 (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin) ↔ (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∧ ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ Fin))
5547, 53, 54sylanbrc 697 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ (𝒫 𝑦 ∩ Fin)) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin))
56 eqid 2621 . . . . . . . . . . . . . . . . 17 𝐽 = 𝐽
5756topopn 20636 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → 𝐽𝐽)
58 difexg 4768 . . . . . . . . . . . . . . . 16 ( 𝐽𝐽 → ( 𝐽𝑟) ∈ V)
5957, 58syl 17 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top → ( 𝐽𝑟) ∈ V)
6059ralrimivw 2961 . . . . . . . . . . . . . 14 (𝐽 ∈ Top → ∀𝑟𝑦 ( 𝐽𝑟) ∈ V)
61 eqid 2621 . . . . . . . . . . . . . . 15 (𝑟𝑦 ↦ ( 𝐽𝑟)) = (𝑟𝑦 ↦ ( 𝐽𝑟))
6261fnmpt 5977 . . . . . . . . . . . . . 14 (∀𝑟𝑦 ( 𝐽𝑟) ∈ V → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
6360, 62syl 17 . . . . . . . . . . . . 13 (𝐽 ∈ Top → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
6463ad3antrrr 765 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → (𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦)
65 simpr 477 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin))
66 elfpw 8212 . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
6945ad2antrr 761 . . . . . . . . . . . . 13 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
7068, 69sseqtrd 3620 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
7167simprd 479 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → 𝑤 ∈ Fin)
72 fipreima 8216 . . . . . . . . . . . 12 (((𝑟𝑦 ↦ ( 𝐽𝑟)) Fn 𝑦𝑤 ⊆ ran (𝑟𝑦 ↦ ( 𝐽𝑟)) ∧ 𝑤 ∈ Fin) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤)
7364, 70, 71, 72syl3anc 1323 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)) → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤)
74 eqcom 2628 . . . . . . . . . . . 12 (((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = 𝑤𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7574rexbii 3034 . . . . . . . . . . 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 477 . . . . . . . . . . . 12 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7877inteqd 4445 . . . . . . . . . . 11 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
7978eqeq2d 2631 . . . . . . . . . 10 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑤 = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) → (∅ = 𝑤 ↔ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
8055, 76, 79rexxfrd 4841 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
81 0ex 4750 . . . . . . . . . 10 ∅ ∈ V
82 imassrn 5436 . . . . . . . . . . . . 13 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ⊆ ran (𝑟𝐽 ↦ ( 𝐽𝑟))
83 eqid 2621 . . . . . . . . . . . . . . . . 17 (𝑟𝐽 ↦ ( 𝐽𝑟)) = (𝑟𝐽 ↦ ( 𝐽𝑟))
8456, 83opncldf1 20798 . . . . . . . . . . . . . . . 16 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽1-1-onto→(Clsd‘𝐽) ∧ (𝑟𝐽 ↦ ( 𝐽𝑟)) = (𝑣 ∈ (Clsd‘𝐽) ↦ ( 𝐽𝑣))))
8584simpld 475 . . . . . . . . . . . . . . 15 (𝐽 ∈ Top → (𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽1-1-onto→(Clsd‘𝐽))
86 f1ofo 6101 . . . . . . . . . . . . . . 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 6075 . . . . . . . . . . . . . 14 ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽) → ran (𝑟𝐽 ↦ ( 𝐽𝑟)) = (Clsd‘𝐽))
8987, 88syl 17 . . . . . . . . . . . . 13 (𝐽 ∈ Top → ran (𝑟𝐽 ↦ ( 𝐽𝑟)) = (Clsd‘𝐽))
9082, 89syl5sseq 3632 . . . . . . . . . . . 12 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ⊆ (Clsd‘𝐽))
91 fvex 6158 . . . . . . . . . . . . 13 (Clsd‘𝐽) ∈ V
9291elpw2 4788 . . . . . . . . . . . 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 761 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
95 elfi 8263 . . . . . . . . . 10 ((∅ ∈ V ∧ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽)) → (∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤))
9681, 94, 95sylancr 694 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ ∃𝑤 ∈ (𝒫 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∩ Fin)∅ = 𝑤))
97 inundif 4018 . . . . . . . . . . . . . 14 (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) = (𝒫 𝑦 ∩ Fin)
9897rexeqi 3132 . . . . . . . . . . . . 13 (∃𝑧 ∈ (((𝒫 𝑦 ∩ Fin) ∩ {∅}) ∪ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) 𝐽 = 𝑧 ↔ ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)
99 rexun 3771 . . . . . . . . . . . . 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 3812 . . . . . . . . . . . . . . . . . . . . . 22 ((𝒫 𝑦 ∩ Fin) ∩ {∅}) ⊆ {∅}
102101sseli 3579 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 ∈ {∅})
103 elsni 4165 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ {∅} → 𝑧 = ∅)
104102, 103syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
105104unieqd 4412 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
106 uni0 4431 . . . . . . . . . . . . . . . . . . 19 ∅ = ∅
107105, 106syl6eq 2671 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → 𝑧 = ∅)
108107eqeq2d 2631 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ( 𝐽 = 𝑧 𝐽 = ∅))
109108biimpd 219 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) → ( 𝐽 = 𝑧 𝐽 = ∅))
110109rexlimiv 3020 . . . . . . . . . . . . . . 15 (∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∩ {∅}) 𝐽 = 𝑧 𝐽 = ∅)
111 ssid 3603 . . . . . . . . . . . . . . . . . . . 20 𝑦𝑦
112111a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦𝑦)
113 simprr 795 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 = ∅)
114 0ss 3944 . . . . . . . . . . . . . . . . . . . . . . . . 25 ∅ ⊆ 𝑦
115113, 114syl6eqss 3634 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 𝑦)
11625ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 𝐽)
117115, 116eqssd 3600 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝐽 = 𝑦)
118117, 113eqtr3d 2657 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 = ∅)
119118, 3syl6eqel 2706 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ Fin)
120 pwfi 8205 . . . . . . . . . . . . . . . . . . . . 21 ( 𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
121119, 120sylib 208 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝒫 𝑦 ∈ Fin)
122 pwuni 4859 . . . . . . . . . . . . . . . . . . . 20 𝑦 ⊆ 𝒫 𝑦
123 ssfi 8124 . . . . . . . . . . . . . . . . . . . 20 ((𝒫 𝑦 ∈ Fin ∧ 𝑦 ⊆ 𝒫 𝑦) → 𝑦 ∈ Fin)
124121, 122, 123sylancl 693 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ Fin)
125 elfpw 8212 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ↔ (𝑦𝑦𝑦 ∈ Fin))
126112, 124, 125sylanbrc 697 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ (𝒫 𝑦 ∩ Fin))
127 simprl 793 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ≠ ∅)
128 eldifsn 4287 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑦 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑦 ≠ ∅))
129126, 127, 128sylanbrc 697 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → 𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}))
130 unieq 4410 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑦 𝑧 = 𝑦)
131130eqeq2d 2631 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑦 → ( 𝐽 = 𝑧 𝐽 = 𝑦))
132131rspcev 3295 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ 𝐽 = 𝑦) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧)
133129, 117, 132syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ (𝑦 ≠ ∅ ∧ 𝐽 = ∅)) → ∃𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) 𝐽 = 𝑧)
134133expr 642 . . . . . . . . . . . . . . 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 395 . . . . . . . . . . . . 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 399 . . . . . . . . . . . . 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 3710 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ∈ (𝒫 𝑦 ∩ Fin))
142141adantl 482 . . . . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧𝑦)
145 simpllr 798 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑦𝐽)
146144, 145sstrd 3593 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧𝐽)
147146unissd 4428 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 𝐽)
148 eqss 3598 . . . . . . . . . . . . . . . 16 ( 𝑧 = 𝐽 ↔ ( 𝑧 𝐽 𝐽 𝑧))
149148baib 943 . . . . . . . . . . . . . . 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 2628 . . . . . . . . . . . . . 14 ( 𝑧 = 𝐽 𝐽 = 𝑧)
152 ssdif0 3916 . . . . . . . . . . . . . . 15 ( 𝐽 𝑧 ↔ ( 𝐽 𝑧) = ∅)
153 eqcom 2628 . . . . . . . . . . . . . . 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 5087 . . . . . . . . . . . . . . . . . 18 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧)
157144resmptd 5411 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧) = (𝑟𝑧 ↦ ( 𝐽𝑟)))
158157rneqd 5313 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ran ((𝑟𝑦 ↦ ( 𝐽𝑟)) ↾ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
159156, 158syl5eq 2667 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
160159inteqd 4445 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ran (𝑟𝑧 ↦ ( 𝐽𝑟)))
16159ralrimivw 2961 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ Top → ∀𝑟𝑧 ( 𝐽𝑟) ∈ V)
162161ad3antrrr 765 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ∀𝑟𝑧 ( 𝐽𝑟) ∈ V)
163 dfiin3g 5339 . . . . . . . . . . . . . . . . 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 4287 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ↔ (𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ 𝑧 ≠ ∅))
166165simprbi 480 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) → 𝑧 ≠ ∅)
167166adantl 482 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → 𝑧 ≠ ∅)
168 iindif2 4555 . . . . . . . . . . . . . . . . 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 2661 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ( 𝐽 𝑟𝑧 𝑟))
171 uniiun 4539 . . . . . . . . . . . . . . . 16 𝑧 = 𝑟𝑧 𝑟
172171difeq2i 3703 . . . . . . . . . . . . . . 15 ( 𝐽 𝑧) = ( 𝐽 𝑟𝑧 𝑟)
173170, 172syl6eqr 2673 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) ∧ 𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅})) → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ( 𝐽 𝑧))
174173eqeq2d 2631 . . . . . . . . . . . . 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 3042 . . . . . . . . . . 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 5421 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ ∅))
179 ima0 5440 . . . . . . . . . . . . . . . . . . . 20 ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ ∅) = ∅
180178, 179syl6eq 2671 . . . . . . . . . . . . . . . . . . 19 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ∅)
181180inteqd 4445 . . . . . . . . . . . . . . . . . 18 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = ∅)
182 int0 4455 . . . . . . . . . . . . . . . . . 18 ∅ = V
183181, 182syl6eq 2671 . . . . . . . . . . . . . . . . 17 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) = V)
184183neeq1d 2849 . . . . . . . . . . . . . . . 16 (𝑧 = ∅ → ( ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ≠ ∅ ↔ V ≠ ∅))
18515, 184mpbiri 248 . . . . . . . . . . . . . . 15 (𝑧 = ∅ → ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) ≠ ∅)
186185necomd 2845 . . . . . . . . . . . . . 14 (𝑧 = ∅ → ∅ ≠ ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧))
187186necon2i 2824 . . . . . . . . . . . . 13 (∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧) → 𝑧 ≠ ∅)
188165rbaibr 945 . . . . . . . . . . . . 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 669 . . . . . . . . . . 11 ((𝑧 ∈ (𝒫 𝑦 ∩ Fin) ∧ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)) ↔ (𝑧 ∈ ((𝒫 𝑦 ∩ Fin) ∖ {∅}) ∧ ∅ = ((𝑟𝑦 ↦ ( 𝐽𝑟)) “ 𝑧)))
191190rexbii2 3032 . . . . . . . . . 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 334 . . . . . . 7 (((𝐽 ∈ Top ∧ 𝑦𝐽) ∧ 𝑦 ≠ ∅) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
19524, 194pm2.61dane 2877 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑦𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ ( 𝑟𝑦 ( 𝐽𝑟) = ∅ → ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))))
19660adantr 481 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ∀𝑟𝑦 ( 𝐽𝑟) ∈ V)
197 dfiin3g 5339 . . . . . . . . . . 11 (∀𝑟𝑦 ( 𝐽𝑟) ∈ V → 𝑟𝑦 ( 𝐽𝑟) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
198196, 197syl 17 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑟𝑦 ( 𝐽𝑟) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
19945inteqd 4445 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ran (𝑟𝑦 ↦ ( 𝐽𝑟)))
200198, 199eqtr4d 2658 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽) → 𝑟𝑦 ( 𝐽𝑟) = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
201200eqeq1d 2623 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ∅))
202 nne 2794 . . . . . . . 8 ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) = ∅)
203201, 202syl6bbr 278 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝑟𝑦 ( 𝐽𝑟) = ∅ ↔ ¬ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅))
204203imbi1d 331 . . . . . 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 348 . . . . 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 491 . . 3 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → (( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
209208ralbidva 2979 . 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 21101 . . 3 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
211210baib 943 . 2 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑦 ∈ 𝒫 𝐽( 𝐽 = 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin) 𝐽 = 𝑧)))
21293adantr 481 . . 3 ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝒫 𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ∈ 𝒫 (Clsd‘𝐽))
213 simpl 473 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝐽 ∈ Top)
214 funmpt 5884 . . . . . 6 Fun (𝑟𝐽 ↦ ( 𝐽𝑟))
215214a1i 11 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → Fun (𝑟𝐽 ↦ ( 𝐽𝑟)))
216 elpwi 4140 . . . . . . 7 (𝑥 ∈ 𝒫 (Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽))
217 foima 6077 . . . . . . . . 9 ((𝑟𝐽 ↦ ( 𝐽𝑟)):𝐽onto→(Clsd‘𝐽) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽) = (Clsd‘𝐽))
21887, 217syl 17 . . . . . . . 8 (𝐽 ∈ Top → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽) = (Clsd‘𝐽))
219218sseq2d 3612 . . . . . . 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 445 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → 𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽))
222 ssimaexg 6221 . . . . 5 ((𝐽 ∈ Top ∧ Fun (𝑟𝐽 ↦ ( 𝐽𝑟)) ∧ 𝑥 ⊆ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝐽)) → ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
223213, 215, 221, 222syl3anc 1323 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → ∃𝑦(𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
224 df-rex 2913 . . . . 5 (∃𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ↔ ∃𝑦(𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
225 selpw 4137 . . . . . . 7 (𝑦 ∈ 𝒫 𝐽𝑦𝐽)
226225anbi1i 730 . . . . . 6 ((𝑦 ∈ 𝒫 𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) ↔ (𝑦𝐽𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
227226exbii 1771 . . . . 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 477 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
231230fveq2d 6152 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (fi‘𝑥) = (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)))
232231eleq2d 2684 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (∅ ∈ (fi‘𝑥) ↔ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
233232notbid 308 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → (¬ ∅ ∈ (fi‘𝑥) ↔ ¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))))
234230inteqd 4445 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦))
235234neeq1d 2849 . . . 4 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ( 𝑥 ≠ ∅ ↔ ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅))
236233, 235imbi12d 334 . . 3 ((𝐽 ∈ Top ∧ 𝑥 = ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((¬ ∅ ∈ (fi‘𝑥) → 𝑥 ≠ ∅) ↔ (¬ ∅ ∈ (fi‘((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦)) → ((𝑟𝐽 ↦ ( 𝐽𝑟)) “ 𝑦) ≠ ∅)))
237212, 229, 236ralxfrd 4839 . 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 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cun 3553  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   cuni 4402   cint 4440   ciun 4485   ciin 4486  cmpt 4673  ccnv 5073  ran crn 5075  cres 5076  cima 5077  Fun wfun 5841   Fn wfn 5842  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847  Fincfn 7899  ficfi 8260  Topctop 20617  Clsdccld 20730  Compccmp 21099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fi 8261  df-top 20621  df-cld 20733  df-cmp 21100
This theorem is referenced by:  cmpfii  21122  fclscmp  21744  heibor1lem  33240
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