Theorem elrelimasn 5128

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 5105	. . . . . 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 3609	. . . . 5 |- ( E. y e. { A } y R B -> E. y y e. { A } )
4		elsni 3707	. . . . . 6 |- ( y e. { A } -> y = A )
5	4	eximi 1649	. . . . 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 2820	. . . 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 4789	. . 3 |- ( ( Rel R /\ A R B ) -> A e. _V )
11	10	ex 115	. 2 |- ( Rel R -> ( A R B -> A e. _V ) )
12		imasng 5127	. . . . 5 |- ( A e. _V -> ( R " { A } ) = { x | A R x } )
13	12	eleq2d 2302	. . . 4 |- ( A e. _V -> ( B e. ( R " { A } ) <-> B e. { x | A R x } ) )
14		brrelex2 4791	. . . . . 6 |- ( ( Rel R /\ A R B ) -> B e. _V )
15	14	ex 115	. . . . 5 |- ( Rel R -> ( A R B -> B e. _V ) )
16		breq2 4113	. . . . . 6 |- ( x = B -> ( A R x <-> A R B ) )
17	16	elab3g 2968	. . . . 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 713	1 |- ( Rel R -> ( B e. ( R " { A } ) <-> A R B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1398   E.wex 1541   e. wcel 2203   {cab 2218   E.wrex 2521   _Vcvv 2813   {csn 3689   class class class wbr 4109   "cima 4752   Rel wrel 4754

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762

This theorem is referenced by:  eliniseg2  5142