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Theorem frmin 9786
Description: Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6369 and tz7.5 6406. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.)
Assertion
Ref Expression
frmin (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Distinct variable groups:   𝑦,𝐵   𝑦,𝑅
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem frmin
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frss 5652 . . . 4 (𝐵𝐴 → (𝑅 Fr 𝐴𝑅 Fr 𝐵))
2 sess2 5654 . . . 4 (𝐵𝐴 → (𝑅 Se 𝐴𝑅 Se 𝐵))
31, 2anim12d 609 . . 3 (𝐵𝐴 → ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝑅 Fr 𝐵𝑅 Se 𝐵)))
4 n0 4358 . . . 4 (𝐵 ≠ ∅ ↔ ∃𝑏 𝑏𝐵)
5 predeq3 6326 . . . . . . . . . . 11 (𝑦 = 𝑏 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑏))
65eqeq1d 2736 . . . . . . . . . 10 (𝑦 = 𝑏 → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, 𝐵, 𝑏) = ∅))
76rspcev 3621 . . . . . . . . 9 ((𝑏𝐵 ∧ Pred(𝑅, 𝐵, 𝑏) = ∅) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
87ex 412 . . . . . . . 8 (𝑏𝐵 → (Pred(𝑅, 𝐵, 𝑏) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
98adantl 481 . . . . . . 7 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → (Pred(𝑅, 𝐵, 𝑏) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
10 predres 6361 . . . . . . . . . . 11 Pred(𝑅, 𝐵, 𝑏) = Pred((𝑅𝐵), 𝐵, 𝑏)
11 relres 6025 . . . . . . . . . . . . 13 Rel (𝑅𝐵)
12 ssttrcl 9752 . . . . . . . . . . . . 13 (Rel (𝑅𝐵) → (𝑅𝐵) ⊆ t++(𝑅𝐵))
1311, 12ax-mp 5 . . . . . . . . . . . 12 (𝑅𝐵) ⊆ t++(𝑅𝐵)
14 predrelss 6359 . . . . . . . . . . . 12 ((𝑅𝐵) ⊆ t++(𝑅𝐵) → Pred((𝑅𝐵), 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
1513, 14ax-mp 5 . . . . . . . . . . 11 Pred((𝑅𝐵), 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏)
1610, 15eqsstri 4029 . . . . . . . . . 10 Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏)
17 ssn0 4409 . . . . . . . . . 10 ((Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ Pred(𝑅, 𝐵, 𝑏) ≠ ∅) → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅)
1816, 17mpan 690 . . . . . . . . 9 (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅)
19 predss 6330 . . . . . . . . 9 Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵
2018, 19jctil 519 . . . . . . . 8 (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → (Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅))
21 dffr4 6342 . . . . . . . . . . . 12 (𝑅 Fr 𝐵 ↔ ∀𝑐((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅))
2221biimpi 216 . . . . . . . . . . 11 (𝑅 Fr 𝐵 → ∀𝑐((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅))
23 ttrclse 9764 . . . . . . . . . . . . 13 (𝑅 Se 𝐵 → t++(𝑅𝐵) Se 𝐵)
24 setlikespec 6347 . . . . . . . . . . . . 13 ((𝑏𝐵 ∧ t++(𝑅𝐵) Se 𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V)
2523, 24sylan2 593 . . . . . . . . . . . 12 ((𝑏𝐵𝑅 Se 𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V)
2625ancoms 458 . . . . . . . . . . 11 ((𝑅 Se 𝐵𝑏𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V)
27 sseq1 4020 . . . . . . . . . . . . . . 15 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (𝑐𝐵 ↔ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵))
28 neeq1 3000 . . . . . . . . . . . . . . 15 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (𝑐 ≠ ∅ ↔ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅))
2927, 28anbi12d 632 . . . . . . . . . . . . . 14 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → ((𝑐𝐵𝑐 ≠ ∅) ↔ (Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅)))
30 predeq2 6325 . . . . . . . . . . . . . . . 16 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → Pred(𝑅, 𝑐, 𝑦) = Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦))
3130eqeq1d 2736 . . . . . . . . . . . . . . 15 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
3231rexeqbi1dv 3336 . . . . . . . . . . . . . 14 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
3329, 32imbi12d 344 . . . . . . . . . . . . 13 (𝑐 = Pred(t++(𝑅𝐵), 𝐵, 𝑏) → (((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) ↔ ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅)))
3433spcgv 3595 . . . . . . . . . . . 12 (Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V → (∀𝑐((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅)))
3534impcom 407 . . . . . . . . . . 11 ((∀𝑐((𝑐𝐵𝑐 ≠ ∅) → ∃𝑦𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∈ V) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
3622, 26, 35syl2an 596 . . . . . . . . . 10 ((𝑅 Fr 𝐵 ∧ (𝑅 Se 𝐵𝑏𝐵)) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
3736anassrs 467 . . . . . . . . 9 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
38 predres 6361 . . . . . . . . . . . . . . . . 17 Pred(𝑅, 𝐵, 𝑦) = Pred((𝑅𝐵), 𝐵, 𝑦)
39 predrelss 6359 . . . . . . . . . . . . . . . . . 18 ((𝑅𝐵) ⊆ t++(𝑅𝐵) → Pred((𝑅𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑦))
4013, 39ax-mp 5 . . . . . . . . . . . . . . . . 17 Pred((𝑅𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑦)
4138, 40eqsstri 4029 . . . . . . . . . . . . . . . 16 Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑦)
42 inss1 4244 . . . . . . . . . . . . . . . . . . . 20 (t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅𝐵)
43 coss1 5868 . . . . . . . . . . . . . . . . . . . 20 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅𝐵) → ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))))
4442, 43ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵)))
45 coss2 5869 . . . . . . . . . . . . . . . . . . . 20 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅𝐵) → (t++(𝑅𝐵) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ t++(𝑅𝐵)))
4642, 45ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (t++(𝑅𝐵) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ t++(𝑅𝐵))
4744, 46sstri 4004 . . . . . . . . . . . . . . . . . 18 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅𝐵) ∘ t++(𝑅𝐵))
48 ttrcltr 9753 . . . . . . . . . . . . . . . . . 18 (t++(𝑅𝐵) ∘ t++(𝑅𝐵)) ⊆ t++(𝑅𝐵)
4947, 48sstri 4004 . . . . . . . . . . . . . . . . 17 ((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅𝐵)
50 predtrss 6344 . . . . . . . . . . . . . . . . 17 ((((t++(𝑅𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅𝐵) ∧ 𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
5149, 50mp3an1 1447 . . . . . . . . . . . . . . . 16 ((𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(t++(𝑅𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
5241, 51sstrid 4006 . . . . . . . . . . . . . . 15 ((𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏))
53 sspred 6331 . . . . . . . . . . . . . . 15 ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅𝐵), 𝐵, 𝑏)) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦))
5419, 52, 53sylancr 587 . . . . . . . . . . . . . 14 ((𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ∧ 𝑏𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦))
5554ancoms 458 . . . . . . . . . . . . 13 ((𝑏𝐵𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏)) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦))
5655eqeq1d 2736 . . . . . . . . . . . 12 ((𝑏𝐵𝑦 ∈ Pred(t++(𝑅𝐵), 𝐵, 𝑏)) → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
5756rexbidva 3174 . . . . . . . . . . 11 (𝑏𝐵 → (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ ∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅))
58 ssrexv 4064 . . . . . . . . . . . 12 (Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 → (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
5919, 58ax-mp 5 . . . . . . . . . . 11 (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
6057, 59biimtrrdi 254 . . . . . . . . . 10 (𝑏𝐵 → (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
6160adantl 481 . . . . . . . . 9 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → (∃𝑦 ∈ Pred (t++(𝑅𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅𝐵), 𝐵, 𝑏), 𝑦) = ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
6237, 61syld 47 . . . . . . . 8 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → ((Pred(t++(𝑅𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
6320, 62syl5 34 . . . . . . 7 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
649, 63pm2.61dne 3025 . . . . . 6 (((𝑅 Fr 𝐵𝑅 Se 𝐵) ∧ 𝑏𝐵) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
6564ex 412 . . . . 5 ((𝑅 Fr 𝐵𝑅 Se 𝐵) → (𝑏𝐵 → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
6665exlimdv 1930 . . . 4 ((𝑅 Fr 𝐵𝑅 Se 𝐵) → (∃𝑏 𝑏𝐵 → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
674, 66biimtrid 242 . . 3 ((𝑅 Fr 𝐵𝑅 Se 𝐵) → (𝐵 ≠ ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
683, 67syl6com 37 . 2 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝐵𝐴 → (𝐵 ≠ ∅ → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)))
6968imp32 418 1 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐵𝐴𝐵 ≠ ∅)) → ∃𝑦𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1534   = wceq 1536  wex 1775  wcel 2105  wne 2937  wrex 3067  Vcvv 3477  cin 3961  wss 3962  c0 4338   Fr wfr 5637   Se wse 5638   × cxp 5686  cres 5690  ccom 5692  Rel wrel 5693  Predcpred 6321  t++cttrcl 9744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-ttrcl 9745
This theorem is referenced by:  frind  9787  frr1  9796
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