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Theorem prodex 14618
 Description: A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)
Assertion
Ref Expression
prodex 𝑘𝐴 𝐵 ∈ V

Proof of Theorem prodex
Dummy variables 𝑓 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 14617 . 2 𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
2 iotaex 5856 . 2 (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))) ∈ V
31, 2eqeltri 2695 1 𝑘𝐴 𝐵 ∈ V
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 383   ∧ wa 384   ∧ w3a 1036   = wceq 1481  ∃wex 1702   ∈ wcel 1988   ≠ wne 2791  ∃wrex 2910  Vcvv 3195  ⦋csb 3526   ⊆ wss 3567  ifcif 4077   class class class wbr 4644   ↦ cmpt 4720  ℩cio 5837  –1-1-onto→wf1o 5875  ‘cfv 5876  (class class class)co 6635  0cc0 9921  1c1 9922   · cmul 9926  ℕcn 11005  ℤcz 11362  ℤ≥cuz 11672  ...cfz 12311  seqcseq 12784   ⇝ cli 14196  ∏cprod 14616 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-sn 4169  df-pr 4171  df-uni 4428  df-iota 5839  df-prod 14617 This theorem is referenced by:  risefacval  14720  fallfacval  14721  prmoval  15718  fprodsubrecnncnvlem  39884  fprodaddrecnncnvlem  39886  etransclem13  40227  ovnlecvr  40535  ovncvrrp  40541  hoidmvval  40554  vonioolem1  40657
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