Theorem releldmb 4571

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 4530	. . 3 |- ( A e. dom R -> ( A e. dom R <-> E. x A R x ) )
2	1	ibi 165	. 2 |- ( A e. dom R -> E. x A R x )
3		releldm 4569	. . . 4 |- ( ( Rel R /\ A R x ) -> A e. dom R )
4	3	ex 108	. . 3 |- ( Rel R -> ( A R x -> A e. dom R ) )
5	4	exlimdv 1700	. 2 |- ( Rel R -> ( E. x A R x -> A e. dom R ) )
6	2, 5	impbid2 131	1 |- ( Rel R -> ( A e. dom R <-> E. x A R x ) )

Syntax hints:   -> wi 4   <-> wb 98   E.wex 1381   e. wcel 1393   class class class wbr 3764   dom cdm 4345   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355

This theorem is referenced by: (None)