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Theorem ixpm 6632
 Description: If an infinite Cartesian product of a family 𝐵(𝑥) is inhabited, every 𝐵(𝑥) is inhabited. (Contributed by Mario Carneiro, 22-Jun-2016.) (Revised by Jim Kingdon, 16-Feb-2023.)
Assertion
Ref Expression
ixpm (∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴𝑧 𝑧𝐵)
Distinct variable groups:   𝐴,𝑓   𝑧,𝑓,𝐵   𝑥,𝑓,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑧)   𝐵(𝑥)

Proof of Theorem ixpm
StepHypRef Expression
1 df-ixp 6601 . . . 4 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
21abeq2i 2251 . . 3 (𝑓X𝑥𝐴 𝐵 ↔ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵))
3 elex2 2705 . . . 4 ((𝑓𝑥) ∈ 𝐵 → ∃𝑧 𝑧𝐵)
43ralimi 2498 . . 3 (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴𝑧 𝑧𝐵)
52, 4simplbiim 385 . 2 (𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴𝑧 𝑧𝐵)
65exlimiv 1578 1 (∃𝑓 𝑓X𝑥𝐴 𝐵 → ∀𝑥𝐴𝑧 𝑧𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∀wral 2417   Fn wfn 5126  ‘cfv 5131  Xcixp 6600 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-v 2691  df-ixp 6601 This theorem is referenced by:  ixp0  6633
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