Theorem elreimasng 4970

Description: Elementhood in the image of a singleton. (Contributed by Jim Kingdon, 10-Dec-2018.)

Assertion

Ref	Expression
elreimasng	|- ( ( Rel R /\ A e. V ) -> ( B e. ( R " { A } ) <-> A R B ) )

Proof of Theorem elreimasng

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasng 4969	. . 3 |- ( A e. V -> ( R " { A } ) = { x | A R x } )
2	1	eleq2d 2236	. 2 |- ( A e. V -> ( B e. ( R " { A } ) <-> B e. { x | A R x } ) )
3		brrelex2 4645	. . . 4 |- ( ( Rel R /\ A R B ) -> B e. _V )
4	3	ex 114	. . 3 |- ( Rel R -> ( A R B -> B e. _V ) )
5		breq2 3986	. . . 4 |- ( x = B -> ( A R x <-> A R B ) )
6	5	elab3g 2877	. . 3 |- ( ( A R B -> B e. _V ) -> ( B e. { x | A R x } <-> A R B ) )
7	4, 6	syl 14	. 2 |- ( Rel R -> ( B e. { x | A R x } <-> A R B ) )
8	2, 7	sylan9bbr 459	1 |- ( ( Rel R /\ A e. V ) -> ( B e. ( R " { A } ) <-> A R B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2136   {cab 2151   _Vcvv 2726   {csn 3576   class class class wbr 3982   "cima 4607   Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by: (None)