Theorem ixpm 6978

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

Step	Hyp	Ref	Expression
1		df-ixp 6947	. . . 4 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
2	1	abeq2i 2345	. . 3 |- ( f e. X_ x e. A B <-> ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) )
3		elex2 2832	. . . 4 |- ( ( f ` x ) e. B -> E. z z e. B )
4	3	ralimi 2607	. . 3 |- ( A. x e. A ( f ` x ) e. B -> A. x e. A E. z z e. B )
5	2, 4	simplbiim 387	. 2 |- ( f e. X_ x e. A B -> A. x e. A E. z z e. B )
6	5	exlimiv 1647	1 |- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )

Syntax hints:   -> wi 4   /\ wa 104   E.wex 1541   e. wcel 2205   {cab 2220   A.wral 2522   Fn wfn 5352   ` cfv 5357   X_cixp 6946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-ral 2527  df-v 2817  df-ixp 6947

This theorem is referenced by:  ixp0  6979