| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > prodex | Structured version Visualization version GIF version | ||
| Description: A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.) |
| Ref | Expression |
|---|---|
| prodex | ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-prod 15940 | . 2 ⊢ ∏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)))) | |
| 2 | iotaex 6534 | . 2 ⊢ (℩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)))) ∈ V | |
| 3 | 1, 2 | eqeltri 2837 | 1 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 Vcvv 3480 ⦋csb 3899 ⊆ wss 3951 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 ℩cio 6512 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 · cmul 11160 ℕcn 12266 ℤcz 12613 ℤ≥cuz 12878 ...cfz 13547 seqcseq 14042 ⇝ cli 15520 ∏cprod 15939 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-sn 4627 df-pr 4629 df-uni 4908 df-iota 6514 df-prod 15940 |
| This theorem is referenced by: risefacval 16044 fallfacval 16045 prmoval 17071 fprodsubrecnncnvlem 45922 fprodaddrecnncnvlem 45924 etransclem13 46262 ovnlecvr 46573 ovncvrrp 46579 hoidmvval 46592 vonioolem1 46695 |
| Copyright terms: Public domain | W3C validator |