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Theorem prodex 15941
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 15940 . 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 6534 . 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 2837 1 𝑘𝐴 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-ontowf1o 6560  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   · cmul 11160  cn 12266  cz 12613  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
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