Theorem elrelimasn 4995

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 4975	. . . . . 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 3523	. . . . 5 |- ( E. y e. { A } y R B -> E. y y e. { A } )
4		elsni 3611	. . . . . 6 |- ( y e. { A } -> y = A )
5	4	eximi 1600	. . . . 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 2744	. . . 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 4666	. . 3 |- ( ( Rel R /\ A R B ) -> A e. _V )
11	10	ex 115	. 2 |- ( Rel R -> ( A R B -> A e. _V ) )
12		imasng 4994	. . . . 5 |- ( A e. _V -> ( R " { A } ) = { x | A R x } )
13	12	eleq2d 2247	. . . 4 |- ( A e. _V -> ( B e. ( R " { A } ) <-> B e. { x | A R x } ) )
14		brrelex2 4668	. . . . . 6 |- ( ( Rel R /\ A R B ) -> B e. _V )
15	14	ex 115	. . . . 5 |- ( Rel R -> ( A R B -> B e. _V ) )
16		breq2 4008	. . . . . 6 |- ( x = B -> ( A R x <-> A R B ) )
17	16	elab3g 2889	. . . . 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 705	1 |- ( Rel R -> ( B e. ( R " { A } ) <-> A R B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   {cab 2163   E.wrex 2456   _Vcvv 2738   {csn 3593   class class class wbr 4004   "cima 4630   Rel wrel 4632

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-br 4005  df-opab 4066  df-xp 4633  df-rel 4634  df-cnv 4635  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640

This theorem is referenced by:  eliniseg2  5009