Proof of Theorem seqsval
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | df-seqs 28290 | . . 3
⊢
seqs𝑀(
+ , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉),
〈𝑀, (𝐹‘𝑀)〉) “ ω) | 
| 2 |  | eqid 2737 | . . . . . 6
⊢ V =
V | 
| 3 |  | fvoveq1 7454 | . . . . . . . . . 10
⊢ (𝑧 = 𝑥 → (𝐹‘(𝑧 +s 1s )) = (𝐹‘(𝑥 +s 1s
))) | 
| 4 | 3 | oveq2d 7447 | . . . . . . . . 9
⊢ (𝑧 = 𝑥 → (𝑤 + (𝐹‘(𝑧 +s 1s ))) = (𝑤 + (𝐹‘(𝑥 +s 1s
)))) | 
| 5 |  | oveq1 7438 | . . . . . . . . 9
⊢ (𝑤 = 𝑦 → (𝑤 + (𝐹‘(𝑥 +s 1s ))) = (𝑦 + (𝐹‘(𝑥 +s 1s
)))) | 
| 6 |  | eqid 2737 | . . . . . . . . 9
⊢ (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s )))) = (𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s
)))) | 
| 7 |  | ovex 7464 | . . . . . . . . 9
⊢ (𝑦 + (𝐹‘(𝑥 +s 1s ))) ∈
V | 
| 8 | 4, 5, 6, 7 | ovmpo 7593 | . . . . . . . 8
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦) = (𝑦 + (𝐹‘(𝑥 +s 1s
)))) | 
| 9 | 8 | el2v 3487 | . . . . . . 7
⊢ (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦) = (𝑦 + (𝐹‘(𝑥 +s 1s
))) | 
| 10 | 9 | opeq2i 4877 | . . . . . 6
⊢
〈(𝑥
+s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉 = 〈(𝑥 +s 1s ),
(𝑦 + (𝐹‘(𝑥 +s 1s
)))〉 | 
| 11 | 2, 2, 10 | mpoeq123i 7509 | . . . . 5
⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉) = (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s
)))〉) | 
| 12 |  | rdgeq1 8451 | . . . . 5
⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉) = (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉)
→ rec((𝑥 ∈ V,
𝑦 ∈ V ↦
〈(𝑥 +s
1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉),
〈𝑀, (𝐹‘𝑀)〉)) | 
| 13 | 11, 12 | ax-mp 5 | . . . 4
⊢
rec((𝑥 ∈ V,
𝑦 ∈ V ↦
〈(𝑥 +s
1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉),
〈𝑀, (𝐹‘𝑀)〉) | 
| 14 | 13 | imaeq1i 6075 | . . 3
⊢
(rec((𝑥 ∈ V,
𝑦 ∈ V ↦
〈(𝑥 +s
1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉),
〈𝑀, (𝐹‘𝑀)〉) “ ω) | 
| 15 |  | df-ima 5698 | . . 3
⊢
(rec((𝑥 ∈ V,
𝑦 ∈ V ↦
〈(𝑥 +s
1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) “ ω) = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) ↾ ω) | 
| 16 | 1, 14, 15 | 3eqtr2i 2771 | . 2
⊢
seqs𝑀(
+ , 𝐹) = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) ↾ ω) | 
| 17 |  | seqsval.1 | . . 3
⊢ (𝜑 → 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) ↾ ω)) | 
| 18 | 17 | rneqd 5949 | . 2
⊢ (𝜑 → ran 𝑅 = ran (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑥(𝑧 ∈ V, 𝑤 ∈ V ↦ (𝑤 + (𝐹‘(𝑧 +s 1s ))))𝑦)〉), 〈𝑀, (𝐹‘𝑀)〉) ↾ ω)) | 
| 19 | 16, 18 | eqtr4id 2796 | 1
⊢ (𝜑 → seqs𝑀( + , 𝐹) = ran 𝑅) |