Detailed syntax breakdown of Definition df-prod
| Step | Hyp | Ref
| Expression |
| 1 | | cA |
. . 3
class 𝐴 |
| 2 | | cB |
. . 3
class 𝐵 |
| 3 | | vk |
. . 3
setvar 𝑘 |
| 4 | 1, 2, 3 | cprod 15939 |
. 2
class
∏𝑘 ∈
𝐴 𝐵 |
| 5 | | vm |
. . . . . . . . 9
setvar 𝑚 |
| 6 | 5 | cv 1539 |
. . . . . . . 8
class 𝑚 |
| 7 | | cuz 12878 |
. . . . . . . 8
class
ℤ≥ |
| 8 | 6, 7 | cfv 6561 |
. . . . . . 7
class
(ℤ≥‘𝑚) |
| 9 | 1, 8 | wss 3951 |
. . . . . 6
wff 𝐴 ⊆
(ℤ≥‘𝑚) |
| 10 | | vy |
. . . . . . . . . . 11
setvar 𝑦 |
| 11 | 10 | cv 1539 |
. . . . . . . . . 10
class 𝑦 |
| 12 | | cc0 11155 |
. . . . . . . . . 10
class
0 |
| 13 | 11, 12 | wne 2940 |
. . . . . . . . 9
wff 𝑦 ≠ 0 |
| 14 | | cmul 11160 |
. . . . . . . . . . 11
class
· |
| 15 | | cz 12613 |
. . . . . . . . . . . 12
class
ℤ |
| 16 | 3 | cv 1539 |
. . . . . . . . . . . . . 14
class 𝑘 |
| 17 | 16, 1 | wcel 2108 |
. . . . . . . . . . . . 13
wff 𝑘 ∈ 𝐴 |
| 18 | | c1 11156 |
. . . . . . . . . . . . 13
class
1 |
| 19 | 17, 2, 18 | cif 4525 |
. . . . . . . . . . . 12
class if(𝑘 ∈ 𝐴, 𝐵, 1) |
| 20 | 3, 15, 19 | cmpt 5225 |
. . . . . . . . . . 11
class (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) |
| 21 | | vn |
. . . . . . . . . . . 12
setvar 𝑛 |
| 22 | 21 | cv 1539 |
. . . . . . . . . . 11
class 𝑛 |
| 23 | 14, 20, 22 | cseq 14042 |
. . . . . . . . . 10
class seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) |
| 24 | | cli 15520 |
. . . . . . . . . 10
class
⇝ |
| 25 | 23, 11, 24 | wbr 5143 |
. . . . . . . . 9
wff seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦 |
| 26 | 13, 25 | wa 395 |
. . . . . . . 8
wff (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) |
| 27 | 26, 10 | wex 1779 |
. . . . . . 7
wff
∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) |
| 28 | 27, 21, 8 | wrex 3070 |
. . . . . 6
wff
∃𝑛 ∈
(ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) |
| 29 | 14, 20, 6 | cseq 14042 |
. . . . . . 7
class seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) |
| 30 | | vx |
. . . . . . . 8
setvar 𝑥 |
| 31 | 30 | cv 1539 |
. . . . . . 7
class 𝑥 |
| 32 | 29, 31, 24 | wbr 5143 |
. . . . . 6
wff seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 |
| 33 | 9, 28, 32 | w3a 1087 |
. . . . 5
wff (𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) |
| 34 | 33, 5, 15 | wrex 3070 |
. . . 4
wff
∃𝑚 ∈
ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) |
| 35 | | cfz 13547 |
. . . . . . . . 9
class
... |
| 36 | 18, 6, 35 | co 7431 |
. . . . . . . 8
class
(1...𝑚) |
| 37 | | vf |
. . . . . . . . 9
setvar 𝑓 |
| 38 | 37 | cv 1539 |
. . . . . . . 8
class 𝑓 |
| 39 | 36, 1, 38 | wf1o 6560 |
. . . . . . 7
wff 𝑓:(1...𝑚)–1-1-onto→𝐴 |
| 40 | | cn 12266 |
. . . . . . . . . . 11
class
ℕ |
| 41 | 22, 38 | cfv 6561 |
. . . . . . . . . . . 12
class (𝑓‘𝑛) |
| 42 | 3, 41, 2 | csb 3899 |
. . . . . . . . . . 11
class
⦋(𝑓‘𝑛) / 𝑘⦌𝐵 |
| 43 | 21, 40, 42 | cmpt 5225 |
. . . . . . . . . 10
class (𝑛 ∈ ℕ ↦
⦋(𝑓‘𝑛) / 𝑘⦌𝐵) |
| 44 | 14, 43, 18 | cseq 14042 |
. . . . . . . . 9
class seq1(
· , (𝑛 ∈
ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)) |
| 45 | 6, 44 | cfv 6561 |
. . . . . . . 8
class (seq1(
· , (𝑛 ∈
ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) |
| 46 | 31, 45 | wceq 1540 |
. . . . . . 7
wff 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦
⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) |
| 47 | 39, 46 | wa 395 |
. . . . . 6
wff (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) |
| 48 | 47, 37 | wex 1779 |
. . . . 5
wff
∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) |
| 49 | 48, 5, 40 | wrex 3070 |
. . . 4
wff
∃𝑚 ∈
ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) |
| 50 | 34, 49 | wo 848 |
. . 3
wff
(∃𝑚 ∈
ℤ (𝐴 ⊆
(ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) |
| 51 | 50, 30 | cio 6512 |
. 2
class
(℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈
(ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) |
| 52 | 4, 51 | wceq 1540 |
1
wff
∏𝑘 ∈
𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈
(ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) |