Theorem releldmb 4784

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

Assertion

Ref	Expression
releldmb	|- ( Rel R -> ( A e. dom R <-> E. x A R x ) )

Distinct variable groups:   x, A   x, R

Proof of Theorem releldmb

Step	Hyp	Ref	Expression
1		eldmg 4742	. . 3 |- ( A e. dom R -> ( A e. dom R <-> E. x A R x ) )
2	1	ibi 175	. 2 |- ( A e. dom R -> E. x A R x )
3		releldm 4782	. . . 4 |- ( ( Rel R /\ A R x ) -> A e. dom R )
4	3	ex 114	. . 3 |- ( Rel R -> ( A R x -> A e. dom R ) )
5	4	exlimdv 1792	. 2 |- ( Rel R -> ( E. x A R x -> A e. dom R ) )
6	2, 5	impbid2 142	1 |- ( Rel R -> ( A e. dom R <-> E. x A R x ) )

Syntax hints:   -> wi 4   <-> wb 104   E.wex 1469   e. wcel 1481   class class class wbr 3937   dom cdm 4547   Rel wrel 4552

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-dm 4557

This theorem is referenced by: (None)