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Theorem trpredrec 32052
Description: If 𝑌 is an 𝑅, 𝐴 transitive predecessor, then it is either an immediate predecessor or there is a transitive predecessor between 𝑌 and 𝑋. (Contributed by Scott Fenton, 9-May-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
trpredrec ((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋   𝑧,𝑌

Proof of Theorem trpredrec
Dummy variables 𝑎 𝑖 𝑗 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eltrpred 32040 . 2 (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) ↔ ∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖))
2 nn0suc 7314 . . . 4 (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
3 fveq2 6402 . . . . . . . . . . 11 (𝑖 = ∅ → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅))
43eleq2d 2867 . . . . . . . . . 10 (𝑖 = ∅ → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ↔ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅)))
54anbi2d 616 . . . . . . . . 9 (𝑖 = ∅ → (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) ↔ ((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅))))
65biimpd 220 . . . . . . . 8 (𝑖 = ∅ → (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅))))
7 setlikespec 5908 . . . . . . . . . . 11 ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
8 fr0g 7761 . . . . . . . . . . 11 (Pred(𝑅, 𝐴, 𝑋) ∈ V → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
97, 8syl 17 . . . . . . . . . 10 ((𝑋𝐴𝑅 Se 𝐴) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
109eleq2d 2867 . . . . . . . . 9 ((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
1110biimpa 464 . . . . . . . 8 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅)) → 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋))
126, 11syl6com 37 . . . . . . 7 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → (𝑖 = ∅ → 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
13 fveq2 6402 . . . . . . . . . . . . 13 (𝑖 = suc 𝑗 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗))
1413eleq2d 2867 . . . . . . . . . . . 12 (𝑖 = suc 𝑗 → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ↔ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗)))
1514anbi2d 616 . . . . . . . . . . 11 (𝑖 = suc 𝑗 → (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) ↔ ((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗))))
1615biimpd 220 . . . . . . . . . 10 (𝑖 = suc 𝑗 → (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗))))
17 fvex 6415 . . . . . . . . . . . . . . . . 17 ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ∈ V
18 trpredlem1 32041 . . . . . . . . . . . . . . . . . . . . 21 (Pred(𝑅, 𝐴, 𝑋) ∈ V → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐴)
197, 18syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝑋𝐴𝑅 Se 𝐴) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐴)
2019sseld 3791 . . . . . . . . . . . . . . . . . . 19 ((𝑋𝐴𝑅 Se 𝐴) → (𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → 𝑧𝐴))
21 setlikespec 5908 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
2221expcom 400 . . . . . . . . . . . . . . . . . . . 20 (𝑅 Se 𝐴 → (𝑧𝐴 → Pred(𝑅, 𝐴, 𝑧) ∈ V))
2322adantl 469 . . . . . . . . . . . . . . . . . . 19 ((𝑋𝐴𝑅 Se 𝐴) → (𝑧𝐴 → Pred(𝑅, 𝐴, 𝑧) ∈ V))
2420, 23syld 47 . . . . . . . . . . . . . . . . . 18 ((𝑋𝐴𝑅 Se 𝐴) → (𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → Pred(𝑅, 𝐴, 𝑧) ∈ V))
2524ralrimiv 3149 . . . . . . . . . . . . . . . . 17 ((𝑋𝐴𝑅 Se 𝐴) → ∀𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V)
26 iunexg 7367 . . . . . . . . . . . . . . . . 17 ((((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ∈ V ∧ ∀𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V) → 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V)
2717, 25, 26sylancr 577 . . . . . . . . . . . . . . . 16 ((𝑋𝐴𝑅 Se 𝐴) → 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V)
28 nfcv 2944 . . . . . . . . . . . . . . . . 17 𝑎Pred(𝑅, 𝐴, 𝑋)
29 nfcv 2944 . . . . . . . . . . . . . . . . 17 𝑎𝑗
30 nfmpt1 4934 . . . . . . . . . . . . . . . . . . . . 21 𝑎(𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦))
3130, 28nfrdg 7740 . . . . . . . . . . . . . . . . . . . 20 𝑎rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
32 nfcv 2944 . . . . . . . . . . . . . . . . . . . 20 𝑎ω
3331, 32nfres 5593 . . . . . . . . . . . . . . . . . . 19 𝑎(rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
3433, 29nffv 6412 . . . . . . . . . . . . . . . . . 18 𝑎((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
35 nfcv 2944 . . . . . . . . . . . . . . . . . 18 𝑎Pred(𝑅, 𝐴, 𝑧)
3634, 35nfiun 4733 . . . . . . . . . . . . . . . . 17 𝑎 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)
37 eqid 2802 . . . . . . . . . . . . . . . . 17 (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
38 predeq3 5891 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
3938cbviunv 4744 . . . . . . . . . . . . . . . . . 18 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑧𝑎 Pred(𝑅, 𝐴, 𝑧)
40 iuneq1 4719 . . . . . . . . . . . . . . . . . 18 (𝑎 = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → 𝑧𝑎 Pred(𝑅, 𝐴, 𝑧) = 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧))
4139, 40syl5eq 2848 . . . . . . . . . . . . . . . . 17 (𝑎 = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧))
4228, 29, 36, 37, 41frsucmpt 7763 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ω ∧ 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ∈ V) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧))
4327, 42sylan2 582 . . . . . . . . . . . . . . 15 ((𝑗 ∈ ω ∧ (𝑋𝐴𝑅 Se 𝐴)) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) = 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧))
4443eleq2d 2867 . . . . . . . . . . . . . 14 ((𝑗 ∈ ω ∧ (𝑋𝐴𝑅 Se 𝐴)) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) ↔ 𝑌 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)))
4544biimpd 220 . . . . . . . . . . . . 13 ((𝑗 ∈ ω ∧ (𝑋𝐴𝑅 Se 𝐴)) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗) → 𝑌 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)))
4645expimpd 443 . . . . . . . . . . . 12 (𝑗 ∈ ω → (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗)) → 𝑌 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)))
47 eliun 4709 . . . . . . . . . . . . 13 (𝑌 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) ↔ ∃𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)𝑌 ∈ Pred(𝑅, 𝐴, 𝑧))
48 ssiun2 4748 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
49 dftrpred2 32033 . . . . . . . . . . . . . . . . . 18 TrPred(𝑅, 𝐴, 𝑋) = 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
5048, 49syl6sseqr 3843 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ TrPred(𝑅, 𝐴, 𝑋))
5150sseld 3791 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ω → (𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → 𝑧 ∈ TrPred(𝑅, 𝐴, 𝑋)))
52 vex 3390 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
5352elpredim 5899 . . . . . . . . . . . . . . . . 17 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → 𝑌𝑅𝑧)
5453a1i 11 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ω → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → 𝑌𝑅𝑧))
5551, 54anim12d 598 . . . . . . . . . . . . . . 15 (𝑗 ∈ ω → ((𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ∧ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑧)) → (𝑧 ∈ TrPred(𝑅, 𝐴, 𝑋) ∧ 𝑌𝑅𝑧)))
5655reximdv2 3197 . . . . . . . . . . . . . 14 (𝑗 ∈ ω → (∃𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
5756com12 32 . . . . . . . . . . . . 13 (∃𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
5847, 57sylbi 208 . . . . . . . . . . . 12 (𝑌 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧) → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
5946, 58syl6com 37 . . . . . . . . . . 11 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗)) → (𝑗 ∈ ω → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
6059pm2.43d 53 . . . . . . . . . 10 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑗)) → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
6116, 60syl6com 37 . . . . . . . . 9 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → (𝑖 = suc 𝑗 → (𝑗 ∈ ω → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
6261com23 86 . . . . . . . 8 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → (𝑗 ∈ ω → (𝑖 = suc 𝑗 → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
6362rexlimdv 3214 . . . . . . 7 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → (∃𝑗 ∈ ω 𝑖 = suc 𝑗 → ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))
6412, 63orim12d 978 . . . . . 6 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
6564ex 399 . . . . 5 ((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))))
6665com23 86 . . . 4 ((𝑋𝐴𝑅 Se 𝐴) → ((𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))))
672, 66syl5 34 . . 3 ((𝑋𝐴𝑅 Se 𝐴) → (𝑖 ∈ ω → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧))))
6867rexlimdv 3214 . 2 ((𝑋𝐴𝑅 Se 𝐴) → (∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
691, 68syl5bi 233 1 ((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ∨ ∃𝑧 ∈ TrPred (𝑅, 𝐴, 𝑋)𝑌𝑅𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 865   = wceq 1637  wcel 2155  wral 3092  wrex 3093  Vcvv 3387  wss 3763  c0 4110   ciun 4705   class class class wbr 4837  cmpt 4916   Se wse 5262  cres 5307  Predcpred 5886  suc csuc 5932  cfv 6095  ωcom 7289  reccrdg 7735  TrPredctrpred 32031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-se 5265  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-om 7290  df-wrecs 7636  df-recs 7698  df-rdg 7736  df-trpred 32032
This theorem is referenced by: (None)
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