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Theorem trpredtr 31854
 Description: The transitive predecessors are transitive in 𝑅 and 𝐴 (Contributed by Scott Fenton, 20-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
trpredtr ((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))

Proof of Theorem trpredtr
Dummy variables 𝑎 𝑓 𝑖 𝑗 𝑡 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eltrpred 31850 . 2 (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) ↔ ∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖))
2 simplr 807 . . . . . 6 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → 𝑖 ∈ ω)
3 peano2 7128 . . . . . 6 (𝑖 ∈ ω → suc 𝑖 ∈ ω)
42, 3syl 17 . . . . 5 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → suc 𝑖 ∈ ω)
5 simpr 476 . . . . . . 7 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 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)‘𝑖))
6 ssid 3657 . . . . . . 7 Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)
7 predeq3 5722 . . . . . . . . . 10 (𝑡 = 𝑌 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑌))
87sseq2d 3666 . . . . . . . . 9 (𝑡 = 𝑌 → (Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑡) ↔ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)))
98rspcev 3340 . . . . . . . 8 ((𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)) → ∃𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑡))
10 ssiun 4594 . . . . . . . 8 (∃𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑡) → Pred(𝑅, 𝐴, 𝑌) ⊆ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
119, 10syl 17 . . . . . . 7 ((𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ Pred(𝑅, 𝐴, 𝑌)) → Pred(𝑅, 𝐴, 𝑌) ⊆ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
125, 6, 11sylancl 695 . . . . . 6 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
13 fvex 6239 . . . . . . . 8 ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∈ V
14 setlikespec 5739 . . . . . . . . . . . . 13 ((𝑋𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
15 trpredlem1 31851 . . . . . . . . . . . . 13 (Pred(𝑅, 𝐴, 𝑋) ∈ V → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
1614, 15syl 17 . . . . . . . . . . . 12 ((𝑋𝐴𝑅 Se 𝐴) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐴)
1716sseld 3635 . . . . . . . . . . 11 ((𝑋𝐴𝑅 Se 𝐴) → (𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → 𝑡𝐴))
18 setlikespec 5739 . . . . . . . . . . . . 13 ((𝑡𝐴𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑡) ∈ V)
1918expcom 450 . . . . . . . . . . . 12 (𝑅 Se 𝐴 → (𝑡𝐴 → Pred(𝑅, 𝐴, 𝑡) ∈ V))
2019adantl 481 . . . . . . . . . . 11 ((𝑋𝐴𝑅 Se 𝐴) → (𝑡𝐴 → Pred(𝑅, 𝐴, 𝑡) ∈ V))
2117, 20syld 47 . . . . . . . . . 10 ((𝑋𝐴𝑅 Se 𝐴) → (𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → Pred(𝑅, 𝐴, 𝑡) ∈ V))
2221ralrimiv 2994 . . . . . . . . 9 ((𝑋𝐴𝑅 Se 𝐴) → ∀𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V)
2322ad2antrr 762 . . . . . . . 8 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → ∀𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V)
24 iunexg 7185 . . . . . . . 8 ((((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ∈ V ∧ ∀𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V) → 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V)
2513, 23, 24sylancr 696 . . . . . . 7 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V)
26 nfcv 2793 . . . . . . . 8 𝑓Pred(𝑅, 𝐴, 𝑋)
27 nfcv 2793 . . . . . . . 8 𝑓𝑖
28 nfcv 2793 . . . . . . . 8 𝑓 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡)
29 predeq3 5722 . . . . . . . . . . . 12 (𝑦 = 𝑡 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑡))
3029cbviunv 4591 . . . . . . . . . . 11 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑡𝑎 Pred(𝑅, 𝐴, 𝑡)
31 iuneq1 4566 . . . . . . . . . . 11 (𝑎 = 𝑓 𝑡𝑎 Pred(𝑅, 𝐴, 𝑡) = 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡))
3230, 31syl5eq 2697 . . . . . . . . . 10 (𝑎 = 𝑓 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦) = 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡))
3332cbvmptv 4783 . . . . . . . . 9 (𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ∈ V ↦ 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡))
34 rdgeq1 7552 . . . . . . . . 9 ((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ∈ V ↦ 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡)) → rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑓 ∈ V ↦ 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡)), Pred(𝑅, 𝐴, 𝑋)))
35 reseq1 5422 . . . . . . . . 9 (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑓 ∈ V ↦ 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡)), Pred(𝑅, 𝐴, 𝑋)) → (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑓 ∈ V ↦ 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω))
3633, 34, 35mp2b 10 . . . . . . . 8 (rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑓 ∈ V ↦ 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
37 iuneq1 4566 . . . . . . . 8 (𝑓 = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → 𝑡𝑓 Pred(𝑅, 𝐴, 𝑡) = 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
3826, 27, 28, 36, 37frsucmpt 7578 . . . . . . 7 ((𝑖 ∈ ω ∧ 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡) ∈ V) → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖) = 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
392, 25, 38syl2anc 694 . . . . . 6 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((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 𝑖) = 𝑡 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)Pred(𝑅, 𝐴, 𝑡))
4012, 39sseqtr4d 3675 . . . . 5 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖))
41 fveq2 6229 . . . . . . . . 9 (𝑗 = suc 𝑖 → ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) = ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖))
4241sseq2d 3666 . . . . . . . 8 (𝑗 = suc 𝑖 → (Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ↔ Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖)))
4342rspcev 3340 . . . . . . 7 ((suc 𝑖 ∈ ω ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖)) → ∃𝑗 ∈ ω Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
44 ssiun 4594 . . . . . . 7 (∃𝑗 ∈ ω Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) → Pred(𝑅, 𝐴, 𝑌) ⊆ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
4543, 44syl 17 . . . . . 6 ((suc 𝑖 ∈ ω ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗))
46 dftrpred2 31843 . . . . . 6 TrPred(𝑅, 𝐴, 𝑋) = 𝑗 ∈ ω ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗)
4745, 46syl6sseqr 3685 . . . . 5 ((suc 𝑖 ∈ ω ∧ Pred(𝑅, 𝐴, 𝑌) ⊆ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋))
484, 40, 47syl2anc 694 . . . 4 ((((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) ∧ 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋))
4948ex 449 . . 3 (((𝑋𝐴𝑅 Se 𝐴) ∧ 𝑖 ∈ ω) → (𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))
5049rexlimdva 3060 . 2 ((𝑋𝐴𝑅 Se 𝐴) → (∃𝑖 ∈ ω 𝑌 ∈ ((rec((𝑎 ∈ V ↦ 𝑦𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))
511, 50syl5bi 232 1 ((𝑋𝐴𝑅 Se 𝐴) → (𝑌 ∈ TrPred(𝑅, 𝐴, 𝑋) → Pred(𝑅, 𝐴, 𝑌) ⊆ TrPred(𝑅, 𝐴, 𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  ∪ ciun 4552   ↦ cmpt 4762   Se wse 5100   ↾ cres 5145  Predcpred 5717  suc csuc 5763  ‘cfv 5926  ωcom 7107  reccrdg 7550  TrPredctrpred 31841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-trpred 31842 This theorem is referenced by:  trpredelss  31856  frmin  31867
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