Theorem elfv 6189

Description: Membership in a function value. (Contributed by NM, 30-Apr-2004.)

Assertion

Ref	Expression
elfv	|- ( A e. ( F ` B ) <-> E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) )

Distinct variable groups:   x, A   x, y, B   x, F, y

Allowed substitution hint:   A( y)

Proof of Theorem elfv

Step	Hyp	Ref	Expression
1		fv2 6186	. . 3 |- ( F ` B ) = U. { x | A. y ( B F y <-> y = x ) }
2	1	eleq2i 2693	. 2 |- ( A e. ( F ` B ) <-> A e. U. { x | A. y ( B F y <-> y = x ) } )
3		eluniab 4447	. 2 |- ( A e. U. { x | A. y ( B F y <-> y = x ) } <-> E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) )
4	2, 3	bitri 264	1 |- ( A e. ( F ` B ) <-> E. x ( A e. x /\ A. y ( B F y <-> y = x ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   {cab 2608   U.cuni 4436   class class class wbr 4653   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sn 4178  df-uni 4437  df-iota 5851  df-fv 5896

This theorem is referenced by:  fv3  6206