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Theorem trpredmintr 9325

Description: The transitive predecessors form the smallest superclass of predecessors closed under taking predecessors. (Contributed by Scott Fenton, 25-Apr-2012.) (Revised by Mario Carneiro, 26-Jun-2015.)

**Assertion**
Ref	Expression
trpredmintr	⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → TrPred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)

Distinct variable groups: 𝑦,𝐴 𝑦,𝐵 𝑦,𝑅 𝑦,𝑋

Proof of Theorem trpredmintr Dummy variables 𝑎 𝑐 𝑑 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftrpred2 9313	. 2 ⊢ TrPred(𝑅, 𝐴, 𝑋) = ∪ 𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖)
2		fveq2 6706	. . . . . . . 8 ⊢ (𝑗 = ∅ → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅))
3	2	sseq1d 3922	. . . . . . 7 ⊢ (𝑗 = ∅ → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐵 ↔ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐵))
4	3	imbi2d 344	. . . . . 6 ⊢ (𝑗 = ∅ → ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐵) ↔ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐵)))
5		fveq2 6706	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘))
6	5	sseq1d 3922	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐵 ↔ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵))
7	6	imbi2d 344	. . . . . 6 ⊢ (𝑗 = 𝑘 → ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐵) ↔ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵)))
8		fveq2 6706	. . . . . . . 8 ⊢ (𝑗 = suc 𝑘 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑘))
9	8	sseq1d 3922	. . . . . . 7 ⊢ (𝑗 = suc 𝑘 → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐵 ↔ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑘) ⊆ 𝐵))
10	9	imbi2d 344	. . . . . 6 ⊢ (𝑗 = suc 𝑘 → ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐵) ↔ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑘) ⊆ 𝐵)))
11		fveq2 6706	. . . . . . . 8 ⊢ (𝑗 = 𝑖 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖))
12	11	sseq1d 3922	. . . . . . 7 ⊢ (𝑗 = 𝑖 → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐵 ↔ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐵))
13	12	imbi2d 344	. . . . . 6 ⊢ (𝑗 = 𝑖 → ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑗) ⊆ 𝐵) ↔ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐵)))
14		setlikespec 6172	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑋) ∈ V)
15		fr0g 8160	. . . . . . . . 9 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
16	14, 15	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
17	16	adantr 484	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) = Pred(𝑅, 𝐴, 𝑋))
18		simprr 773	. . . . . . 7 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)
19	17, 18	eqsstrd 3929	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘∅) ⊆ 𝐵)
20		fvex 6719	. . . . . . . . . . 11 ⊢ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ∈ V
21		trpredlem1 9321	. . . . . . . . . . . . . . . 16 ⊢ (Pred(𝑅, 𝐴, 𝑋) ∈ V → ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐴)
22	14, 21	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐴)
23	22	sseld 3890	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) → 𝑦 ∈ 𝐴))
24		setlikespec 6172	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → Pred(𝑅, 𝐴, 𝑦) ∈ V)
25	24	expcom 417	. . . . . . . . . . . . . . 15 ⊢ (𝑅 Se 𝐴 → (𝑦 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑦) ∈ V))
26	25	adantl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑦 ∈ 𝐴 → Pred(𝑅, 𝐴, 𝑦) ∈ V))
27	23, 26	syld 47	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → (𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) → Pred(𝑅, 𝐴, 𝑦) ∈ V))
28	27	ralrimiv 3097	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ∈ V)
29	28	ad2antrr 726	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵) → ∀𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ∈ V)
30		iunexg 7725	. . . . . . . . . . 11 ⊢ ((((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ∈ V ∧ ∀𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ∈ V) → ∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ∈ V)
31	20, 29, 30	sylancr 590	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵) → ∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ∈ V)
32		nfcv 2900	. . . . . . . . . . 11 ⊢ Ⅎ𝑎Pred(𝑅, 𝐴, 𝑋)
33		nfcv 2900	. . . . . . . . . . 11 ⊢ Ⅎ𝑎𝑘
34		nfcv 2900	. . . . . . . . . . 11 ⊢ Ⅎ𝑎∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦)
35		eqid 2734	. . . . . . . . . . 11 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
36		predeq3 6153	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑑 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑑))
37	36	cbviunv 4939	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑑 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑑)
38		iuneq1 4910	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑐 → ∪ 𝑑 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑑) = ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑))
39	37, 38	syl5eq 2786	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑐 → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑))
40	39	cbvmptv 5147	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑))
41		rdgeq1 8136	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)) = (𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)) → rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)))
42		reseq1 5834	. . . . . . . . . . . . . . 15 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) = rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) → (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω))
43	40, 41, 42	mp2b 10	. . . . . . . . . . . . . 14 ⊢ (rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω) = (rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)
44	43	fveq1i 6707	. . . . . . . . . . . . 13 ⊢ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) = ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)
45	44	eqeq2i 2747	. . . . . . . . . . . 12 ⊢ (𝑎 = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ↔ 𝑎 = ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘))
46		iuneq1 4910	. . . . . . . . . . . 12 ⊢ (𝑎 = ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦))
47	45, 46	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑎 = ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) → ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦) = ∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦))
48	32, 33, 34, 35, 47	frsucmpt 8162	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ ∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ∈ V) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑘) = ∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦))
49	31, 48	sylan2 596	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑘) = ∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦))
50	44	sseq1i 3919	. . . . . . . . . . . 12 ⊢ (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵 ↔ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵)
51	50	anbi2i 626	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵) ↔ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵))
52		nfv 1922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴)
53		nfra1 3133	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵
54		nfv 1922	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵
55	53, 54	nfan 1907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦(∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)
56	52, 55	nfan 1907	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵))
57		nfv 1922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵
58	56, 57	nfan 1907	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵)
59		ssel 3884	. . . . . . . . . . . . . 14 ⊢ (((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵 → (𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) → 𝑦 ∈ 𝐵))
60		rsp 3120	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 → (𝑦 ∈ 𝐵 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
61	60	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → (𝑦 ∈ 𝐵 → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
62	59, 61	sylan9r 512	. . . . . . . . . . . . 13 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵) → (𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) → Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵))
63	58, 62	ralrimi 3130	. . . . . . . . . . . 12 ⊢ ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵) → ∀𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
64	63	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ω ∧ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵)) → ∀𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
65	51, 64	sylan2b 597	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵)) → ∀𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
66		iunss 4944	. . . . . . . . . 10 ⊢ (∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ↔ ∀𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
67	65, 66	sylibr 237	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵)) → ∪ 𝑦 ∈ ((rec((𝑐 ∈ V ↦ ∪ 𝑑 ∈ 𝑐 Pred(𝑅, 𝐴, 𝑑)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘)Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵)
68	49, 67	eqsstrd 3929	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) ∧ ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑘) ⊆ 𝐵)
69	68	exp32 424	. . . . . . 7 ⊢ (𝑘 ∈ ω → (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → (((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵 → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑘) ⊆ 𝐵)))
70	69	a2d 29	. . . . . 6 ⊢ (𝑘 ∈ ω → ((((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑘) ⊆ 𝐵) → (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘suc 𝑘) ⊆ 𝐵)))
71	4, 7, 10, 13, 19, 70	finds 7665	. . . . 5 ⊢ (𝑖 ∈ ω → (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐵))
72	71	com12 32	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → (𝑖 ∈ ω → ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐵))
73	72	ralrimiv 3097	. . 3 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ∀𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐵)
74		iunss 4944	. . 3 ⊢ (∪ 𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐵 ↔ ∀𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐵)
75	73, 74	sylibr 237	. 2 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → ∪ 𝑖 ∈ ω ((rec((𝑎 ∈ V ↦ ∪ 𝑦 ∈ 𝑎 Pred(𝑅, 𝐴, 𝑦)), Pred(𝑅, 𝐴, 𝑋)) ↾ ω)‘𝑖) ⊆ 𝐵)
76	1, 75	eqsstrid 3939	1 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑅 Se 𝐴) ∧ (∀𝑦 ∈ 𝐵 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝐵 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)) → TrPred(𝑅, 𝐴, 𝑋) ⊆ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3054 Vcvv 3401 ⊆ wss 3857 ∅c0 4227 ∪ ciun 4894 ↦ cmpt 5124 Se wse 5496 ↾ cres 5542 Predcpred 6148 suc csuc 6204 ‘cfv 6369 ωcom 7633 reccrdg 8134 TrPredctrpred 9311

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-om 7634 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-trpred 9312

This theorem is referenced by: trpredelss 9327 dftrpred3g 9328 trpredpo 9330

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