Theorem ixpm 6554

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 6523	. . . 4 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
2	1	abeq2i 2210	. . 3 |- ( f e. X_ x e. A B <-> ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) )
3		elex2 2657	. . . 4 |- ( ( f ` x ) e. B -> E. z z e. B )
4	3	ralimi 2454	. . 3 |- ( A. x e. A ( f ` x ) e. B -> A. x e. A E. z z e. B )
5	2, 4	simplbiim 382	. 2 |- ( f e. X_ x e. A B -> A. x e. A E. z z e. B )
6	5	exlimiv 1545	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 1436   e. wcel 1448   {cab 2086   A.wral 2375   Fn wfn 5054   ` cfv 5059   X_cixp 6522

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-ral 2380  df-v 2643  df-ixp 6523

This theorem is referenced by:  ixp0  6555