Theorem eliota 5186

Description: An element of an iota expression. (Contributed by Jim Kingdon, 22-Nov-2024.)

Assertion

Ref	Expression
eliota	|- ( A e. ( iota x ph ) <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )

Distinct variable groups:   ph, y   y, A   x, y

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem eliota

Step	Hyp	Ref	Expression
1		dfiota2 5161	. . 3 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }
2	1	eleq2i 2237	. 2 |- ( A e. ( iota x ph ) <-> A e. U. { y | A. x ( ph <-> x = y ) } )
3		eluniab 3808	. 2 |- ( A e. U. { y | A. x ( ph <-> x = y ) } <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )
4	2, 3	bitri 183	1 |- ( A e. ( iota x ph ) <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1346   E.wex 1485   e. wcel 2141   {cab 2156   U.cuni 3796   iotacio 5158

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-sn 3589  df-uni 3797  df-iota 5160

This theorem is referenced by:  eliotaeu  5187