Theorem eliota 5196

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 5171	. . 3 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }
2	1	eleq2i 2242	. 2 |- ( A e. ( iota x ph ) <-> A e. U. { y | A. x ( ph <-> x = y ) } )
3		eluniab 3817	. 2 |- ( A e. U. { y | A. x ( ph <-> x = y ) } <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )
4	2, 3	bitri 184	1 |- ( A e. ( iota x ph ) <-> E. y ( A e. y /\ A. x ( ph <-> x = y ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1351   E.wex 1490   e. wcel 2146   {cab 2161   U.cuni 3805   iotacio 5168

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-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-rex 2459  df-v 2737  df-sn 3595  df-uni 3806  df-iota 5170

This theorem is referenced by:  eliotaeu  5197