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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  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A. x  e.  A  E. z  z  e.  B )
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  |-  X_ x  e.  A  B  =  { f  |  ( f  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( f `  x )  e.  B
) }
21abeq2i 2251 . . 3  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
f `  x )  e.  B ) )
3 elex2 2705 . . . 4  |-  ( ( f `  x )  e.  B  ->  E. z 
z  e.  B )
43ralimi 2498 . . 3  |-  ( A. x  e.  A  (
f `  x )  e.  B  ->  A. x  e.  A  E. z 
z  e.  B )
52, 4simplbiim 385 . 2  |-  ( f  e.  X_ x  e.  A  B  ->  A. x  e.  A  E. z  z  e.  B )
65exlimiv 1578 1  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A. x  e.  A  E. z  z  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1469    e. wcel 1481   {cab 2126   A.wral 2417    Fn wfn 5126   ` cfv 5131   X_cixp 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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