Theorem relelrnb 4842

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 4795	. . 3 |- ( A e. ran R -> ( A e. ran R <-> E. x x R A ) )
2	1	ibi 175	. 2 |- ( A e. ran R -> E. x x R A )
3		relelrn 4840	. . . 4 |- ( ( Rel R /\ x R A ) -> A e. ran R )
4	3	ex 114	. . 3 |- ( Rel R -> ( x R A -> A e. ran R ) )
5	4	exlimdv 1807	. 2 |- ( Rel R -> ( E. x x R A -> A e. ran R ) )
6	2, 5	impbid2 142	1 |- ( Rel R -> ( A e. ran R <-> E. x x R A ) )

Syntax hints:   -> wi 4   <-> wb 104   E.wex 1480   e. wcel 2136   class class class wbr 3982   ran crn 4605   Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by: (None)