Theorem elrelimasn 5102

Description: Elementhood in the image of a singleton. (Contributed by Mario Carneiro, 3-Nov-2015.)

Assertion

Ref	Expression
elrelimasn	|- ( Rel R -> ( B e. ( R " { A } ) <-> A R B ) )

Proof of Theorem elrelimasn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elimag 5080	. . . . . 6 |- ( B e. ( R " { A } ) -> ( B e. ( R " { A } ) <-> E. y e. { A } y R B ) )
2	1	ibi 176	. . . . 5 |- ( B e. ( R " { A } ) -> E. y e. { A } y R B )
3		rexm 3594	. . . . 5 |- ( E. y e. { A } y R B -> E. y y e. { A } )
4		elsni 3687	. . . . . 6 |- ( y e. { A } -> y = A )
5	4	eximi 1648	. . . . 5 |- ( E. y y e. { A } -> E. y y = A )
6	2, 3, 5	3syl 17	. . . 4 |- ( B e. ( R " { A } ) -> E. y y = A )
7		isset 2809	. . . 4 |- ( A e. _V <-> E. y y = A )
8	6, 7	sylibr 134	. . 3 |- ( B e. ( R " { A } ) -> A e. _V )
9	8	a1i 9	. 2 |- ( Rel R -> ( B e. ( R " { A } ) -> A e. _V ) )
10		brrelex1 4765	. . 3 |- ( ( Rel R /\ A R B ) -> A e. _V )
11	10	ex 115	. 2 |- ( Rel R -> ( A R B -> A e. _V ) )
12		imasng 5101	. . . . 5 |- ( A e. _V -> ( R " { A } ) = { x | A R x } )
13	12	eleq2d 2301	. . . 4 |- ( A e. _V -> ( B e. ( R " { A } ) <-> B e. { x | A R x } ) )
14		brrelex2 4767	. . . . . 6 |- ( ( Rel R /\ A R B ) -> B e. _V )
15	14	ex 115	. . . . 5 |- ( Rel R -> ( A R B -> B e. _V ) )
16		breq2 4092	. . . . . 6 |- ( x = B -> ( A R x <-> A R B ) )
17	16	elab3g 2957	. . . . 5 |- ( ( A R B -> B e. _V ) -> ( B e. { x | A R x } <-> A R B ) )
18	15, 17	syl 14	. . . 4 |- ( Rel R -> ( B e. { x | A R x } <-> A R B ) )
19	13, 18	sylan9bbr 463	. . 3 |- ( ( Rel R /\ A e. _V ) -> ( B e. ( R " { A } ) <-> A R B ) )
20	19	ex 115	. 2 |- ( Rel R -> ( A e. _V -> ( B e. ( R " { A } ) <-> A R B ) ) )
21	9, 11, 20	pm5.21ndd 712	1 |- ( Rel R -> ( B e. ( R " { A } ) <-> A R B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802   {csn 3669   class class class wbr 4088   "cima 4728   Rel wrel 4730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  eliniseg2  5116