Theorem elreimasng 4977

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 4976	. . 3 |- ( A e. V -> ( R " { A } ) = { x | A R x } )
2	1	eleq2d 2240	. 2 |- ( A e. V -> ( B e. ( R " { A } ) <-> B e. { x | A R x } ) )
3		brrelex2 4652	. . . 4 |- ( ( Rel R /\ A R B ) -> B e. _V )
4	3	ex 114	. . 3 |- ( Rel R -> ( A R B -> B e. _V ) )
5		breq2 3993	. . . 4 |- ( x = B -> ( A R x <-> A R B ) )
6	5	elab3g 2881	. . 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 460	1 |- ( ( Rel R /\ A e. V ) -> ( B e. ( R " { A } ) <-> A R B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   {cab 2156   _Vcvv 2730   {csn 3583   class class class wbr 3989   "cima 4614   Rel wrel 4616

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by: (None)