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Theorem fin23lem28 9750
Description: Lemma for fin23 9799. The residual is also one-to-one. This preserves the induction invariant. (Contributed by Stefan O'Rear, 2-Nov-2014.)
Hypotheses
Ref Expression
fin23lem.a 𝑈 = seqω((𝑖 ∈ ω, 𝑢 ∈ V ↦ if(((𝑡𝑖) ∩ 𝑢) = ∅, 𝑢, ((𝑡𝑖) ∩ 𝑢))), ran 𝑡)
fin23lem17.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.b 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
fin23lem.c 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
fin23lem.d 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
fin23lem.e 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
Assertion
Ref Expression
fin23lem28 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
Distinct variable groups:   𝑔,𝑖,𝑡,𝑢,𝑣,𝑥,𝑧,𝑎   𝐹,𝑎,𝑡   𝑤,𝑎,𝑥,𝑧,𝑃   𝑣,𝑎,𝑅,𝑖,𝑢   𝑈,𝑎,𝑖,𝑢,𝑣,𝑧   𝑍,𝑎   𝑔,𝑎
Allowed substitution hints:   𝑃(𝑣,𝑢,𝑡,𝑔,𝑖)   𝑄(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖,𝑎)   𝑅(𝑥,𝑧,𝑤,𝑡,𝑔)   𝑈(𝑥,𝑤,𝑡,𝑔)   𝐹(𝑥,𝑧,𝑤,𝑣,𝑢,𝑔,𝑖)   𝑍(𝑥,𝑧,𝑤,𝑣,𝑢,𝑡,𝑔,𝑖)

Proof of Theorem fin23lem28
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fin23lem.e . . 3 𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))
2 eqif 4503 . . 3 (𝑍 = if(𝑃 ∈ Fin, (𝑡𝑅), ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)) ↔ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))))
31, 2mpbi 231 . 2 ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)))
4 difss 4105 . . . . . . . . 9 (ω ∖ 𝑃) ⊆ ω
5 ominf 8718 . . . . . . . . . 10 ¬ ω ∈ Fin
6 fin23lem.b . . . . . . . . . . . . . 14 𝑃 = {𝑣 ∈ ω ∣ ran 𝑈 ⊆ (𝑡𝑣)}
76ssrab3 4054 . . . . . . . . . . . . 13 𝑃 ⊆ ω
8 undif 4426 . . . . . . . . . . . . 13 (𝑃 ⊆ ω ↔ (𝑃 ∪ (ω ∖ 𝑃)) = ω)
97, 8mpbi 231 . . . . . . . . . . . 12 (𝑃 ∪ (ω ∖ 𝑃)) = ω
10 unfi 8773 . . . . . . . . . . . 12 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → (𝑃 ∪ (ω ∖ 𝑃)) ∈ Fin)
119, 10eqeltrrid 2915 . . . . . . . . . . 11 ((𝑃 ∈ Fin ∧ (ω ∖ 𝑃) ∈ Fin) → ω ∈ Fin)
1211ex 413 . . . . . . . . . 10 (𝑃 ∈ Fin → ((ω ∖ 𝑃) ∈ Fin → ω ∈ Fin))
135, 12mtoi 200 . . . . . . . . 9 (𝑃 ∈ Fin → ¬ (ω ∖ 𝑃) ∈ Fin)
14 fin23lem.d . . . . . . . . . 10 𝑅 = (𝑤 ∈ ω ↦ (𝑥 ∈ (ω ∖ 𝑃)(𝑥 ∩ (ω ∖ 𝑃)) ≈ 𝑤))
1514fin23lem22 9737 . . . . . . . . 9 (((ω ∖ 𝑃) ⊆ ω ∧ ¬ (ω ∖ 𝑃) ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
164, 13, 15sylancr 587 . . . . . . . 8 (𝑃 ∈ Fin → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
1716adantl 482 . . . . . . 7 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1-onto→(ω ∖ 𝑃))
18 f1of1 6607 . . . . . . 7 (𝑅:ω–1-1-onto→(ω ∖ 𝑃) → 𝑅:ω–1-1→(ω ∖ 𝑃))
19 f1ss 6573 . . . . . . . 8 ((𝑅:ω–1-1→(ω ∖ 𝑃) ∧ (ω ∖ 𝑃) ⊆ ω) → 𝑅:ω–1-1→ω)
204, 19mpan2 687 . . . . . . 7 (𝑅:ω–1-1→(ω ∖ 𝑃) → 𝑅:ω–1-1→ω)
2117, 18, 203syl 18 . . . . . 6 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → 𝑅:ω–1-1→ω)
22 f1co 6578 . . . . . 6 ((𝑡:ω–1-1→V ∧ 𝑅:ω–1-1→ω) → (𝑡𝑅):ω–1-1→V)
2321, 22syldan 591 . . . . 5 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑡𝑅):ω–1-1→V)
24 f1eq1 6563 . . . . 5 (𝑍 = (𝑡𝑅) → (𝑍:ω–1-1→V ↔ (𝑡𝑅):ω–1-1→V))
2523, 24syl5ibrcom 248 . . . 4 ((𝑡:ω–1-1→V ∧ 𝑃 ∈ Fin) → (𝑍 = (𝑡𝑅) → 𝑍:ω–1-1→V))
2625impr 455 . . 3 ((𝑡:ω–1-1→V ∧ (𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅))) → 𝑍:ω–1-1→V)
27 fvex 6676 . . . . . . . . . . 11 (𝑡𝑧) ∈ V
2827difexi 5223 . . . . . . . . . 10 ((𝑡𝑧) ∖ ran 𝑈) ∈ V
2928rgenw 3147 . . . . . . . . 9 𝑧𝑃 ((𝑡𝑧) ∖ ran 𝑈) ∈ V
30 eqid 2818 . . . . . . . . . 10 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) = (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))
3130fmpt 6866 . . . . . . . . 9 (∀𝑧𝑃 ((𝑡𝑧) ∖ ran 𝑈) ∈ V ↔ (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃⟶V)
3229, 31mpbi 231 . . . . . . . 8 (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃⟶V
3332a1i 11 . . . . . . 7 (𝑡:ω–1-1→V → (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃⟶V)
34 fveq2 6663 . . . . . . . . . . . . 13 (𝑧 = 𝑎 → (𝑡𝑧) = (𝑡𝑎))
3534difeq1d 4095 . . . . . . . . . . . 12 (𝑧 = 𝑎 → ((𝑡𝑧) ∖ ran 𝑈) = ((𝑡𝑎) ∖ ran 𝑈))
36 fvex 6676 . . . . . . . . . . . . 13 (𝑡𝑎) ∈ V
3736difexi 5223 . . . . . . . . . . . 12 ((𝑡𝑎) ∖ ran 𝑈) ∈ V
3835, 30, 37fvmpt 6761 . . . . . . . . . . 11 (𝑎𝑃 → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑡𝑎) ∖ ran 𝑈))
3938ad2antrl 724 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑡𝑎) ∖ ran 𝑈))
40 fveq2 6663 . . . . . . . . . . . . 13 (𝑧 = 𝑏 → (𝑡𝑧) = (𝑡𝑏))
4140difeq1d 4095 . . . . . . . . . . . 12 (𝑧 = 𝑏 → ((𝑡𝑧) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈))
42 fvex 6676 . . . . . . . . . . . . 13 (𝑡𝑏) ∈ V
4342difexi 5223 . . . . . . . . . . . 12 ((𝑡𝑏) ∖ ran 𝑈) ∈ V
4441, 30, 43fvmpt 6761 . . . . . . . . . . 11 (𝑏𝑃 → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) = ((𝑡𝑏) ∖ ran 𝑈))
4544ad2antll 725 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) = ((𝑡𝑏) ∖ ran 𝑈))
4639, 45eqeq12d 2834 . . . . . . . . 9 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) ↔ ((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈)))
47 uneq2 4130 . . . . . . . . . . 11 (((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈) → ( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)))
48 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑎 → (𝑡𝑣) = (𝑡𝑎))
4948sseq2d 3996 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑎 → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡𝑎)))
5049, 6elrab2 3680 . . . . . . . . . . . . . . 15 (𝑎𝑃 ↔ (𝑎 ∈ ω ∧ ran 𝑈 ⊆ (𝑡𝑎)))
5150simprbi 497 . . . . . . . . . . . . . 14 (𝑎𝑃 ran 𝑈 ⊆ (𝑡𝑎))
5251ad2antrl 724 . . . . . . . . . . . . 13 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ran 𝑈 ⊆ (𝑡𝑎))
53 undif 4426 . . . . . . . . . . . . 13 ( ran 𝑈 ⊆ (𝑡𝑎) ↔ ( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = (𝑡𝑎))
5452, 53sylib 219 . . . . . . . . . . . 12 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = (𝑡𝑎))
55 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑣 = 𝑏 → (𝑡𝑣) = (𝑡𝑏))
5655sseq2d 3996 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑏 → ( ran 𝑈 ⊆ (𝑡𝑣) ↔ ran 𝑈 ⊆ (𝑡𝑏)))
5756, 6elrab2 3680 . . . . . . . . . . . . . . 15 (𝑏𝑃 ↔ (𝑏 ∈ ω ∧ ran 𝑈 ⊆ (𝑡𝑏)))
5857simprbi 497 . . . . . . . . . . . . . 14 (𝑏𝑃 ran 𝑈 ⊆ (𝑡𝑏))
5958ad2antll 725 . . . . . . . . . . . . 13 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ran 𝑈 ⊆ (𝑡𝑏))
60 undif 4426 . . . . . . . . . . . . 13 ( ran 𝑈 ⊆ (𝑡𝑏) ↔ ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)) = (𝑡𝑏))
6159, 60sylib 219 . . . . . . . . . . . 12 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)) = (𝑡𝑏))
6254, 61eqeq12d 2834 . . . . . . . . . . 11 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (( ran 𝑈 ∪ ((𝑡𝑎) ∖ ran 𝑈)) = ( ran 𝑈 ∪ ((𝑡𝑏) ∖ ran 𝑈)) ↔ (𝑡𝑎) = (𝑡𝑏)))
6347, 62syl5ib 245 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈) → (𝑡𝑎) = (𝑡𝑏)))
647sseli 3960 . . . . . . . . . . . 12 (𝑎𝑃𝑎 ∈ ω)
657sseli 3960 . . . . . . . . . . . 12 (𝑏𝑃𝑏 ∈ ω)
6664, 65anim12i 612 . . . . . . . . . . 11 ((𝑎𝑃𝑏𝑃) → (𝑎 ∈ ω ∧ 𝑏 ∈ ω))
67 f1fveq 7011 . . . . . . . . . . 11 ((𝑡:ω–1-1→V ∧ (𝑎 ∈ ω ∧ 𝑏 ∈ ω)) → ((𝑡𝑎) = (𝑡𝑏) ↔ 𝑎 = 𝑏))
6866, 67sylan2 592 . . . . . . . . . 10 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → ((𝑡𝑎) = (𝑡𝑏) ↔ 𝑎 = 𝑏))
6963, 68sylibd 240 . . . . . . . . 9 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (((𝑡𝑎) ∖ ran 𝑈) = ((𝑡𝑏) ∖ ran 𝑈) → 𝑎 = 𝑏))
7046, 69sylbid 241 . . . . . . . 8 ((𝑡:ω–1-1→V ∧ (𝑎𝑃𝑏𝑃)) → (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) → 𝑎 = 𝑏))
7170ralrimivva 3188 . . . . . . 7 (𝑡:ω–1-1→V → ∀𝑎𝑃𝑏𝑃 (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) → 𝑎 = 𝑏))
72 dff13 7004 . . . . . . 7 ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃1-1→V ↔ ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃⟶V ∧ ∀𝑎𝑃𝑏𝑃 (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑎) = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈))‘𝑏) → 𝑎 = 𝑏)))
7333, 71, 72sylanbrc 583 . . . . . 6 (𝑡:ω–1-1→V → (𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃1-1→V)
74 fin23lem.c . . . . . . . . 9 𝑄 = (𝑤 ∈ ω ↦ (𝑥𝑃 (𝑥𝑃) ≈ 𝑤))
7574fin23lem22 9737 . . . . . . . 8 ((𝑃 ⊆ ω ∧ ¬ 𝑃 ∈ Fin) → 𝑄:ω–1-1-onto𝑃)
76 f1of1 6607 . . . . . . . 8 (𝑄:ω–1-1-onto𝑃𝑄:ω–1-1𝑃)
7775, 76syl 17 . . . . . . 7 ((𝑃 ⊆ ω ∧ ¬ 𝑃 ∈ Fin) → 𝑄:ω–1-1𝑃)
787, 77mpan 686 . . . . . 6 𝑃 ∈ Fin → 𝑄:ω–1-1𝑃)
79 f1co 6578 . . . . . 6 (((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)):𝑃1-1→V ∧ 𝑄:ω–1-1𝑃) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄):ω–1-1→V)
8073, 78, 79syl2an 595 . . . . 5 ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄):ω–1-1→V)
81 f1eq1 6563 . . . . 5 (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → (𝑍:ω–1-1→V ↔ ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄):ω–1-1→V))
8280, 81syl5ibrcom 248 . . . 4 ((𝑡:ω–1-1→V ∧ ¬ 𝑃 ∈ Fin) → (𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄) → 𝑍:ω–1-1→V))
8382impr 455 . . 3 ((𝑡:ω–1-1→V ∧ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄))) → 𝑍:ω–1-1→V)
8426, 83jaodan 951 . 2 ((𝑡:ω–1-1→V ∧ ((𝑃 ∈ Fin ∧ 𝑍 = (𝑡𝑅)) ∨ (¬ 𝑃 ∈ Fin ∧ 𝑍 = ((𝑧𝑃 ↦ ((𝑡𝑧) ∖ ran 𝑈)) ∘ 𝑄)))) → 𝑍:ω–1-1→V)
853, 84mpan2 687 1 (𝑡:ω–1-1→V → 𝑍:ω–1-1→V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  {cab 2796  wral 3135  {crab 3139  Vcvv 3492  cdif 3930  cun 3931  cin 3932  wss 3933  c0 4288  ifcif 4463  𝒫 cpw 4535   cuni 4830   cint 4867   class class class wbr 5057  cmpt 5137  ran crn 5549  ccom 5552  suc csuc 6186  wf 6344  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348  crio 7102  (class class class)co 7145  cmpo 7147  ωcom 7569  seqωcseqom 8072  m cmap 8395  cen 8494  Fincfn 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356
This theorem is referenced by:  fin23lem32  9754
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