Theorem elreimasng 4863

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 4862	. . 3 |- ( A e. V -> ( R " { A } ) = { x | A R x } )
2	1	eleq2d 2184	. 2 |- ( A e. V -> ( B e. ( R " { A } ) <-> B e. { x | A R x } ) )
3		brrelex2 4540	. . . 4 |- ( ( Rel R /\ A R B ) -> B e. _V )
4	3	ex 114	. . 3 |- ( Rel R -> ( A R B -> B e. _V ) )
5		breq2 3899	. . . 4 |- ( x = B -> ( A R x <-> A R B ) )
6	5	elab3g 2804	. . 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 456	1 |- ( ( Rel R /\ A e. V ) -> ( B e. ( R " { A } ) <-> A R B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1463   {cab 2101   _Vcvv 2657   {csn 3493   class class class wbr 3895   "cima 4502   Rel wrel 4504

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512

This theorem is referenced by: (None)