Theorem relelrnb 4970

Description: Membership in a range. (Contributed by Mario Carneiro, 5-Nov-2015.)

Assertion

Ref	Expression
relelrnb	|- ( Rel R -> ( A e. ran R <-> E. x x R A ) )

Distinct variable groups:   x, A   x, R

Proof of Theorem relelrnb

Step	Hyp	Ref	Expression
1		elrng 4921	. . 3 |- ( A e. ran R -> ( A e. ran R <-> E. x x R A ) )
2	1	ibi 176	. 2 |- ( A e. ran R -> E. x x R A )
3		relelrn 4968	. . . 4 |- ( ( Rel R /\ x R A ) -> A e. ran R )
4	3	ex 115	. . 3 |- ( Rel R -> ( x R A -> A e. ran R ) )
5	4	exlimdv 1867	. 2 |- ( Rel R -> ( E. x x R A -> A e. ran R ) )
6	2, 5	impbid2 143	1 |- ( Rel R -> ( A e. ran R <-> E. x x R A ) )

Syntax hints:   -> wi 4   <-> wb 105   E.wex 1540   e. wcel 2202   class class class wbr 4088   ran crn 4726   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-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

This theorem is referenced by: (None)