Theorem eliota 5203

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 5178	. . 3 |- ( iota x ph ) = U. { y | A. x ( ph <-> x = y ) }
2	1	eleq2i 2244	. 2 |- ( A e. ( iota x ph ) <-> A e. U. { y | A. x ( ph <-> x = y ) } )
3		eluniab 3821	. 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 1492   e. wcel 2148   {cab 2163   U.cuni 3809   iotacio 5175

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sn 3598  df-uni 3810  df-iota 5177

This theorem is referenced by:  eliotaeu  5204