Theorem prodex 15858

Description: A product is a set. (Contributed by Scott Fenton, 4-Dec-2017.)

Assertion

Ref	Expression
prodex	⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V

Proof of Theorem prodex

Dummy variables 𝑓 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prod 15857	. 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 6516	. 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 2828	1 ⊢ ∏𝑘 ∈ 𝐴 𝐵 ∈ V

Syntax hints:   ∧ wa 395   ∨ wo 844   ∧ w3a 1086   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2939  ∃wrex 3069  Vcvv 3473  ⦋csb 3893   ⊆ wss 3948  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  ℩cio 6493  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7412  0cc0 11116  1c1 11117   · cmul 11121  ℕcn 12219  ℤcz 12565  ℤ_≥cuz 12829  ...cfz 13491  seqcseq 13973   ⇝ cli 15435  ∏cprod 15856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-sn 4629  df-pr 4631  df-uni 4909  df-iota 6495  df-prod 15857

This theorem is referenced by:  risefacval  15959  fallfacval  15960  prmoval  16973  fprodsubrecnncnvlem  45084  fprodaddrecnncnvlem  45086  etransclem13  45424  ovnlecvr  45735  ovncvrrp  45741  hoidmvval  45754  vonioolem1  45857