Theorem ixpm 6830

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 6799	. . . 4 |- X_ x e. A B = { f | ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) }
2	1	abeq2i 2317	. . 3 |- ( f e. X_ x e. A B <-> ( f Fn { x | x e. A } /\ A. x e. A ( f ` x ) e. B ) )
3		elex2 2790	. . . 4 |- ( ( f ` x ) e. B -> E. z z e. B )
4	3	ralimi 2570	. . 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 1622	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 1516   e. wcel 2177   {cab 2192   A.wral 2485   Fn wfn 5275   ` cfv 5280   X_cixp 6798

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 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-ral 2490  df-v 2775  df-ixp 6799

This theorem is referenced by:  ixp0  6831