Theorem relelrnb 4900

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 4853	. . 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 4898	. . . 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 1830	. 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 1503   e. wcel 2164   class class class wbr 4029   ran crn 4660   Rel wrel 4664

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by: (None)