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Theorem sprsymrelfo 43510
Description: The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)
Hypotheses
Ref Expression
sprsymrelf.p 𝑃 = 𝒫 (Pairs‘𝑉)
sprsymrelf.r 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
sprsymrelf.f 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
Assertion
Ref Expression
sprsymrelfo (𝑉𝑊𝐹:𝑃onto𝑅)
Distinct variable groups:   𝑃,𝑝   𝑉,𝑐,𝑥,𝑦   𝑝,𝑐,𝑥,𝑦,𝑟   𝑅,𝑝   𝑉,𝑟,𝑐,𝑥,𝑦   𝑊,𝑐,𝑥,𝑦
Allowed substitution hints:   𝑃(𝑥,𝑦,𝑟,𝑐)   𝑅(𝑥,𝑦,𝑟,𝑐)   𝐹(𝑥,𝑦,𝑟,𝑝,𝑐)   𝑉(𝑝)   𝑊(𝑟,𝑝)

Proof of Theorem sprsymrelfo
Dummy variables 𝑎 𝑏 𝑓 𝑞 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sprsymrelf.p . . . 4 𝑃 = 𝒫 (Pairs‘𝑉)
2 sprsymrelf.r . . . 4 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
3 sprsymrelf.f . . . 4 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
41, 2, 3sprsymrelf 43508 . . 3 𝐹:𝑃𝑅
54a1i 11 . 2 (𝑉𝑊𝐹:𝑃𝑅)
6 breq 5065 . . . . . . . . 9 (𝑟 = 𝑡 → (𝑥𝑟𝑦𝑥𝑡𝑦))
7 breq 5065 . . . . . . . . 9 (𝑟 = 𝑡 → (𝑦𝑟𝑥𝑦𝑡𝑥))
86, 7bibi12d 347 . . . . . . . 8 (𝑟 = 𝑡 → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥𝑡𝑦𝑦𝑡𝑥)))
982ralbidv 3204 . . . . . . 7 (𝑟 = 𝑡 → (∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)))
109, 2elrab2 3687 . . . . . 6 (𝑡𝑅 ↔ (𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)))
11 eqid 2826 . . . . . . . . . . 11 {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}
1211sprsymrelfolem1 43505 . . . . . . . . . 10 {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝒫 (Pairs‘𝑉)
1312, 1eleqtrri 2917 . . . . . . . . 9 {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝑃
1413a1i 11 . . . . . . . 8 (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) ∧ 𝑉𝑊) → {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝑃)
15 rexeq 3412 . . . . . . . . . . 11 (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → (∃𝑐𝑓 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
1615opabbidv 5129 . . . . . . . . . 10 (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
1716eqeq2d 2837 . . . . . . . . 9 (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
1817adantl 482 . . . . . . . 8 ((((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) ∧ 𝑉𝑊) ∧ 𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
19 velpw 4550 . . . . . . . . . 10 (𝑡 ∈ 𝒫 (𝑉 × 𝑉) ↔ 𝑡 ⊆ (𝑉 × 𝑉))
20 xpss 5570 . . . . . . . . . . . . . . . 16 (𝑉 × 𝑉) ⊆ (V × V)
21 sstr2 3978 . . . . . . . . . . . . . . . 16 (𝑡 ⊆ (𝑉 × 𝑉) → ((𝑉 × 𝑉) ⊆ (V × V) → 𝑡 ⊆ (V × V)))
2220, 21mpi 20 . . . . . . . . . . . . . . 15 (𝑡 ⊆ (𝑉 × 𝑉) → 𝑡 ⊆ (V × V))
23 df-rel 5561 . . . . . . . . . . . . . . 15 (Rel 𝑡𝑡 ⊆ (V × V))
2422, 23sylibr 235 . . . . . . . . . . . . . 14 (𝑡 ⊆ (𝑉 × 𝑉) → Rel 𝑡)
2524adantl 482 . . . . . . . . . . . . 13 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → Rel 𝑡)
26 dfrel4v 6046 . . . . . . . . . . . . . 14 (Rel 𝑡𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦})
27 nfv 1908 . . . . . . . . . . . . . . . . . . . 20 𝑥(𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉))
28 nfra1 3224 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)
2927, 28nfan 1893 . . . . . . . . . . . . . . . . . . 19 𝑥((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥))
30 nfv 1908 . . . . . . . . . . . . . . . . . . . 20 𝑦(𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉))
31 nfra2 3233 . . . . . . . . . . . . . . . . . . . 20 𝑦𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)
3230, 31nfan 1893 . . . . . . . . . . . . . . . . . . 19 𝑦((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥))
3311sprsymrelfolem2 43506 . . . . . . . . . . . . . . . . . . . 20 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑥𝑡𝑦 ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
34333expa 1112 . . . . . . . . . . . . . . . . . . 19 (((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑥𝑡𝑦 ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
3529, 32, 34opabbid 5128 . . . . . . . . . . . . . . . . . 18 (((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
3635eqeq2d 2837 . . . . . . . . . . . . . . . . 17 (((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
3736biimpd 230 . . . . . . . . . . . . . . . 16 (((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
3837ex 413 . . . . . . . . . . . . . . 15 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
3938com23 86 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4026, 39syl5bi 243 . . . . . . . . . . . . 13 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → (Rel 𝑡 → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4125, 40mpd 15 . . . . . . . . . . . 12 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
4241expcom 414 . . . . . . . . . . 11 (𝑡 ⊆ (𝑉 × 𝑉) → (𝑉𝑊 → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4342com23 86 . . . . . . . . . 10 (𝑡 ⊆ (𝑉 × 𝑉) → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → (𝑉𝑊𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4419, 43sylbi 218 . . . . . . . . 9 (𝑡 ∈ 𝒫 (𝑉 × 𝑉) → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → (𝑉𝑊𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4544imp31 418 . . . . . . . 8 (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) ∧ 𝑉𝑊) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
4614, 18, 45rspcedvd 3630 . . . . . . 7 (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) ∧ 𝑉𝑊) → ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}})
4746ex 413 . . . . . 6 ((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑉𝑊 → ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}}))
4810, 47sylbi 218 . . . . 5 (𝑡𝑅 → (𝑉𝑊 → ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}}))
4948impcom 408 . . . 4 ((𝑉𝑊𝑡𝑅) → ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}})
501, 2, 3sprsymrelfv 43507 . . . . . . 7 (𝑓𝑃 → (𝐹𝑓) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}})
5150adantl 482 . . . . . 6 (((𝑉𝑊𝑡𝑅) ∧ 𝑓𝑃) → (𝐹𝑓) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}})
5251eqeq2d 2837 . . . . 5 (((𝑉𝑊𝑡𝑅) ∧ 𝑓𝑃) → (𝑡 = (𝐹𝑓) ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}}))
5352rexbidva 3301 . . . 4 ((𝑉𝑊𝑡𝑅) → (∃𝑓𝑃 𝑡 = (𝐹𝑓) ↔ ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}}))
5449, 53mpbird 258 . . 3 ((𝑉𝑊𝑡𝑅) → ∃𝑓𝑃 𝑡 = (𝐹𝑓))
5554ralrimiva 3187 . 2 (𝑉𝑊 → ∀𝑡𝑅𝑓𝑃 𝑡 = (𝐹𝑓))
56 dffo3 6866 . 2 (𝐹:𝑃onto𝑅 ↔ (𝐹:𝑃𝑅 ∧ ∀𝑡𝑅𝑓𝑃 𝑡 = (𝐹𝑓)))
575, 55, 56sylanbrc 583 1 (𝑉𝑊𝐹:𝑃onto𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3143  wrex 3144  {crab 3147  Vcvv 3500  wss 3940  𝒫 cpw 4542  {cpr 4566   class class class wbr 5063  {copab 5125  cmpt 5143   × cxp 5552  Rel wrel 5559  wf 6350  ontowfo 6352  cfv 6354  Pairscspr 43490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-spr 43491
This theorem is referenced by:  sprsymrelf1o  43511
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