Theorem releldmb 4848

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 4806	. . 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 4846	. . . 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 1812	. 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 1485   e. wcel 2141   class class class wbr 3989   dom cdm 4611   Rel wrel 4616

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-dm 4621

This theorem is referenced by: (None)