| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | cantnfvalf.f | . . 3
⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅) | 
| 2 | 1 | fnseqom 8496 | . 2
⊢ 𝐹 Fn ω | 
| 3 |  | nn0suc 7917 | . . . 4
⊢ (𝑥 ∈ ω → (𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦)) | 
| 4 |  | fveq2 6905 | . . . . . . 7
⊢ (𝑥 = ∅ → (𝐹‘𝑥) = (𝐹‘∅)) | 
| 5 |  | 0ex 5306 | . . . . . . . 8
⊢ ∅
∈ V | 
| 6 | 1 | seqom0g 8497 | . . . . . . . 8
⊢ (∅
∈ V → (𝐹‘∅) = ∅) | 
| 7 | 5, 6 | ax-mp 5 | . . . . . . 7
⊢ (𝐹‘∅) =
∅ | 
| 8 | 4, 7 | eqtrdi 2792 | . . . . . 6
⊢ (𝑥 = ∅ → (𝐹‘𝑥) = ∅) | 
| 9 |  | 0elon 6437 | . . . . . 6
⊢ ∅
∈ On | 
| 10 | 8, 9 | eqeltrdi 2848 | . . . . 5
⊢ (𝑥 = ∅ → (𝐹‘𝑥) ∈ On) | 
| 11 | 1 | seqomsuc 8498 | . . . . . . . . 9
⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))(𝐹‘𝑦))) | 
| 12 |  | df-ov 7435 | . . . . . . . . 9
⊢ (𝑦(𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))(𝐹‘𝑦)) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘〈𝑦, (𝐹‘𝑦)〉) | 
| 13 | 11, 12 | eqtrdi 2792 | . . . . . . . 8
⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) = ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘〈𝑦, (𝐹‘𝑦)〉)) | 
| 14 |  | df-ov 7435 | . . . . . . . . . . . 12
⊢ (𝐶 +o 𝐷) = ( +o ‘〈𝐶, 𝐷〉) | 
| 15 |  | fnoa 8547 | . . . . . . . . . . . . . 14
⊢ 
+o Fn (On × On) | 
| 16 |  | oacl 8574 | . . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 +o 𝑦) ∈ On) | 
| 17 | 16 | rgen2 3198 | . . . . . . . . . . . . . 14
⊢
∀𝑥 ∈ On
∀𝑦 ∈ On (𝑥 +o 𝑦) ∈ On | 
| 18 |  | ffnov 7560 | . . . . . . . . . . . . . 14
⊢ (
+o :(On × On)⟶On ↔ ( +o Fn (On ×
On) ∧ ∀𝑥 ∈
On ∀𝑦 ∈ On
(𝑥 +o 𝑦) ∈ On)) | 
| 19 | 15, 17, 18 | mpbir2an 711 | . . . . . . . . . . . . 13
⊢ 
+o :(On × On)⟶On | 
| 20 | 19, 9 | f0cli 7117 | . . . . . . . . . . . 12
⊢ (
+o ‘〈𝐶, 𝐷〉) ∈ On | 
| 21 | 14, 20 | eqeltri 2836 | . . . . . . . . . . 11
⊢ (𝐶 +o 𝐷) ∈ On | 
| 22 | 21 | rgen2w 3065 | . . . . . . . . . 10
⊢
∀𝑘 ∈
𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On | 
| 23 |  | eqid 2736 | . . . . . . . . . . 11
⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)) = (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)) | 
| 24 | 23 | fmpo 8094 | . . . . . . . . . 10
⊢
(∀𝑘 ∈
𝐴 ∀𝑧 ∈ 𝐵 (𝐶 +o 𝐷) ∈ On ↔ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On) | 
| 25 | 22, 24 | mpbi 230 | . . . . . . . . 9
⊢ (𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)):(𝐴 × 𝐵)⟶On | 
| 26 | 25, 9 | f0cli 7117 | . . . . . . . 8
⊢ ((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷))‘〈𝑦, (𝐹‘𝑦)〉) ∈ On | 
| 27 | 13, 26 | eqeltrdi 2848 | . . . . . . 7
⊢ (𝑦 ∈ ω → (𝐹‘suc 𝑦) ∈ On) | 
| 28 |  | fveq2 6905 | . . . . . . . 8
⊢ (𝑥 = suc 𝑦 → (𝐹‘𝑥) = (𝐹‘suc 𝑦)) | 
| 29 | 28 | eleq1d 2825 | . . . . . . 7
⊢ (𝑥 = suc 𝑦 → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘suc 𝑦) ∈ On)) | 
| 30 | 27, 29 | syl5ibrcom 247 | . . . . . 6
⊢ (𝑦 ∈ ω → (𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On)) | 
| 31 | 30 | rexlimiv 3147 | . . . . 5
⊢
(∃𝑦 ∈
ω 𝑥 = suc 𝑦 → (𝐹‘𝑥) ∈ On) | 
| 32 | 10, 31 | jaoi 857 | . . . 4
⊢ ((𝑥 = ∅ ∨ ∃𝑦 ∈ ω 𝑥 = suc 𝑦) → (𝐹‘𝑥) ∈ On) | 
| 33 | 3, 32 | syl 17 | . . 3
⊢ (𝑥 ∈ ω → (𝐹‘𝑥) ∈ On) | 
| 34 | 33 | rgen 3062 | . 2
⊢
∀𝑥 ∈
ω (𝐹‘𝑥) ∈ On | 
| 35 |  | ffnfv 7138 | . 2
⊢ (𝐹:ω⟶On ↔ (𝐹 Fn ω ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ On)) | 
| 36 | 2, 34, 35 | mpbir2an 711 | 1
⊢ 𝐹:ω⟶On |