Theorem eliota 5306

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 5279	. . 3 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }
2	1	eleq2i 2296	. 2 |- ( A e. ( iota x ph ) <-> A e. U. { y | A. x ( ph <-> x = y ) } )
3		eluniab 3900	. 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 1393   E.wex 1538   e. wcel 2200   {cab 2215   U.cuni 3888   iotacio 5276

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-sn 3672  df-uni 3889  df-iota 5278

This theorem is referenced by:  eliotaeu  5307