Theorem elreimasng 4905

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 4904	. . 3 |- ( A e. V -> ( R " { A } ) = { x | A R x } )
2	1	eleq2d 2209	. 2 |- ( A e. V -> ( B e. ( R " { A } ) <-> B e. { x | A R x } ) )
3		brrelex2 4580	. . . 4 |- ( ( Rel R /\ A R B ) -> B e. _V )
4	3	ex 114	. . 3 |- ( Rel R -> ( A R B -> B e. _V ) )
5		breq2 3933	. . . 4 |- ( x = B -> ( A R x <-> A R B ) )
6	5	elab3g 2835	. . 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 458	1 |- ( ( Rel R /\ A e. V ) -> ( B e. ( R " { A } ) <-> A R B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   {cab 2125   _Vcvv 2686   {csn 3527   class class class wbr 3929   "cima 4542   Rel wrel 4544

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552

This theorem is referenced by: (None)