Theorem ixpm 6696

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 6665	. . . 4 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
2	1	abeq2i 2277	. . 3 |- ( f e. X_ x e. A B <-> ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) )
3		elex2 2742	. . . 4 |- ( ( f ` x ) e. B -> E. z z e. B )
4	3	ralimi 2529	. . 3 |- ( A. x e. A ( f ` x ) e. B -> A. x e. A E. z z e. B )
5	2, 4	simplbiim 385	. 2 |- ( f e. X_ x e. A B -> A. x e. A E. z z e. B )
6	5	exlimiv 1586	1 |- ( E. f f e. X_ x e. A B -> A. x e. A E. z z e. B )

Syntax hints:   -> wi 4   /\ wa 103   E.wex 1480   e. wcel 2136   {cab 2151   A.wral 2444   Fn wfn 5183   ` cfv 5188   X_cixp 6664

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 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-v 2728  df-ixp 6665

This theorem is referenced by:  ixp0  6697