Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trpredrec Structured version   Visualization version   GIF version

Theorem trpredrec 30784
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 30772 . 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 6955 . . . 4 (𝑖 ∈ ω → (𝑖 = ∅ ∨ ∃𝑗 ∈ ω 𝑖 = suc 𝑗))
3 fveq2 6084 . . . . . . . . . . 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 2668 . . . . . . . . . 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 735 . . . . . . . . 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 217 . . . . . . . 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 5600 . . . . . . . . . . 11 ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
8 fr0g 7391 . . . . . . . . . . 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 2668 . . . . . . . . 9 ((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ↔ 𝑌 ∈ Pred(𝑅, 𝐴, 𝑋)))
1110biimpa 499 . . . . . . . 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 36 . . . . . . 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 6084 . . . . . . . . . . . . 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 2668 . . . . . . . . . . . 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 735 . . . . . . . . . . 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 217 . . . . . . . . . 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 6094 . . . . . . . . . . . . . . . . 17 ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ∈ V
18 trpredlem1 30773 . . . . . . . . . . . . . . . . . . . . 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 3562 . . . . . . . . . . . . . . . . . . 19 ((𝑋𝐴𝑅 Se 𝐴) → (𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → 𝑧𝐴))
21 setlikespec 5600 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑧) ∈ V)
2221expcom 449 . . . . . . . . . . . . . . . . . . . 20 (𝑅 Se 𝐴 → (𝑧𝐴 → Pred(𝑅, 𝐴, 𝑧) ∈ V))
2322adantl 480 . . . . . . . . . . . . . . . . . . 19 ((𝑋𝐴𝑅 Se 𝐴) → (𝑧𝐴 → Pred(𝑅, 𝐴, 𝑧) ∈ V))
2420, 23syld 45 . . . . . . . . . . . . . . . . . 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 2943 . . . . . . . . . . . . . . . . 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 7008 . . . . . . . . . . . . . . . . 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 693 . . . . . . . . . . . . . . . 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 2746 . . . . . . . . . . . . . . . . 17 𝑎Pred(𝑅, 𝐴, 𝑋)
29 nfcv 2746 . . . . . . . . . . . . . . . . 17 𝑎𝑗
30 nfmpt1 4665 . . . . . . . . . . . . . . . . . . . . 21 𝑎(𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦))
3130, 28nfrdg 7370 . . . . . . . . . . . . . . . . . . . 20 𝑎rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋))
32 nfcv 2746 . . . . . . . . . . . . . . . . . . . 20 𝑎ω
3331, 32nfres 5302 . . . . . . . . . . . . . . . . . . 19 𝑎(rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
3433, 29nffv 6091 . . . . . . . . . . . . . . . . . 18 𝑎((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
35 nfcv 2746 . . . . . . . . . . . . . . . . . 18 𝑎Pred(𝑅, 𝐴, 𝑧)
3634, 35nfiun 4474 . . . . . . . . . . . . . . . . 17 𝑎 𝑧 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)Pred(𝑅, 𝐴, 𝑧)
37 eqid 2605 . . . . . . . . . . . . . . . . 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 5583 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
3938cbviunv 4485 . . . . . . . . . . . . . . . . . 18 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑧𝑎 Pred(𝑅, 𝐴, 𝑧)
40 iuneq1 4460 . . . . . . . . . . . . . . . . . 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 2651 . . . . . . . . . . . . . . . . 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 7393 . . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . . 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 2668 . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . 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 626 . . . . . . . . . . . 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 4450 . . . . . . . . . . . . 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 4489 . . . . . . . . . . . . . . . . . 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 30765 . . . . . . . . . . . . . . . . . 18 TrPred(𝑅, 𝐴, 𝑋) = 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
5048, 49syl6sseqr 3610 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ω → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ TrPred(𝑅, 𝐴, 𝑋))
5150sseld 3562 . . . . . . . . . . . . . . . 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 3171 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
5352elpredim 5591 . . . . . . . . . . . . . . . . 17 (𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → 𝑌𝑅𝑧)
5453a1i 11 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ω → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑧) → 𝑌𝑅𝑧))
5551, 54anim12d 583 . . . . . . . . . . . . . . 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 2992 . . . . . . . . . . . . . 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 205 . . . . . . . . . . . 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 36 . . . . . . . . . . 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 50 . . . . . . . . . 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 36 . . . . . . . . 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 83 . . . . . . . 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 3007 . . . . . . 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 878 . . . . . 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 448 . . . . 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 83 . . . 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 33 . . 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 3007 . 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 230 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  wo 381  wa 382   = wceq 1474  wcel 1975  wral 2891  wrex 2892  Vcvv 3168  wss 3535  c0 3869   ciun 4445   class class class wbr 4573  cmpt 4633   Se wse 4981  cres 5026  Predcpred 5578  suc csuc 5624  cfv 5786  ωcom 6930  reccrdg 7365  TrPredctrpred 30763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-om 6931  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-trpred 30764
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator