Theorem breldmg 4541

Description: Membership of first of a binary relation in a domain. (Contributed by NM, 21-Mar-2007.)

Assertion

Ref	Expression
breldmg	|- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R )

Proof of Theorem breldmg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3768	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 2641	. . . 4 |- ( B e. D -> ( A R B -> E. x A R x ) )
3	2	imp 115	. . 3 |- ( ( B e. D /\ A R B ) -> E. x A R x )
4	3	3adant1 922	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> E. x A R x )
5		eldmg 4530	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
6	5	3ad2ant1 925	. 2 |- ( ( A e. C /\ B e. D /\ A R B ) -> ( A e. dom R <-> E. x A R x ) )
7	4, 6	mpbird 156	1 |- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   <-> wb 98   /\ w3a 885   E.wex 1381   e. wcel 1393   class class class wbr 3764   dom cdm 4345

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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

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-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-dm 4355

This theorem is referenced by:  brelrng  4565  releldm  4569  brtposg  5869  shftfvalg  9419  shftfval  9422