| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | seqseq123d.2 | . . . . . . . 8
⊢ (𝜑 → + = 𝑄) | 
| 2 | 1 | oveqd 7449 | . . . . . . 7
⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐹‘(𝑥 +s 1s
)))) | 
| 3 |  | seqseq123d.3 | . . . . . . . . 9
⊢ (𝜑 → 𝐹 = 𝐺) | 
| 4 | 3 | fveq1d 6907 | . . . . . . . 8
⊢ (𝜑 → (𝐹‘(𝑥 +s 1s )) = (𝐺‘(𝑥 +s 1s
))) | 
| 5 | 4 | oveq2d 7448 | . . . . . . 7
⊢ (𝜑 → (𝑦𝑄(𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s
)))) | 
| 6 | 2, 5 | eqtrd 2776 | . . . . . 6
⊢ (𝜑 → (𝑦 + (𝐹‘(𝑥 +s 1s ))) = (𝑦𝑄(𝐺‘(𝑥 +s 1s
)))) | 
| 7 | 6 | opeq2d 4879 | . . . . 5
⊢ (𝜑 → 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉 =
〈(𝑥 +s
1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s
)))〉) | 
| 8 | 7 | mpoeq3dv 7513 | . . . 4
⊢ (𝜑 → (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉) =
(𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ),
(𝑦𝑄(𝐺‘(𝑥 +s 1s
)))〉)) | 
| 9 |  | seqseq123d.1 | . . . . 5
⊢ (𝜑 → 𝑀 = 𝑁) | 
| 10 | 3, 9 | fveq12d 6912 | . . . . 5
⊢ (𝜑 → (𝐹‘𝑀) = (𝐺‘𝑁)) | 
| 11 | 9, 10 | opeq12d 4880 | . . . 4
⊢ (𝜑 → 〈𝑀, (𝐹‘𝑀)〉 = 〈𝑁, (𝐺‘𝑁)〉) | 
| 12 |  | rdgeq12 8454 | . . . 4
⊢ (((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉) =
(𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ),
(𝑦𝑄(𝐺‘(𝑥 +s 1s )))〉) ∧
〈𝑀, (𝐹‘𝑀)〉 = 〈𝑁, (𝐺‘𝑁)〉) → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉),
〈𝑀, (𝐹‘𝑀)〉) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))〉),
〈𝑁, (𝐺‘𝑁)〉)) | 
| 13 | 8, 11, 12 | syl2anc 584 | . . 3
⊢ (𝜑 → rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉),
〈𝑀, (𝐹‘𝑀)〉) = rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))〉),
〈𝑁, (𝐺‘𝑁)〉)) | 
| 14 | 13 | imaeq1d 6076 | . 2
⊢ (𝜑 → (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉),
〈𝑀, (𝐹‘𝑀)〉) “ ω) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))〉),
〈𝑁, (𝐺‘𝑁)〉) “ ω)) | 
| 15 |  | df-seqs 28291 | . 2
⊢
seqs𝑀(
+ , 𝐹) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦 + (𝐹‘(𝑥 +s 1s )))〉),
〈𝑀, (𝐹‘𝑀)〉) “ ω) | 
| 16 |  | df-seqs 28291 | . 2
⊢
seqs𝑁(𝑄, 𝐺) = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 +s 1s ), (𝑦𝑄(𝐺‘(𝑥 +s 1s )))〉),
〈𝑁, (𝐺‘𝑁)〉) “ ω) | 
| 17 | 14, 15, 16 | 3eqtr4g 2801 | 1
⊢ (𝜑 → seqs𝑀( + , 𝐹) = seqs𝑁(𝑄, 𝐺)) |