Theorem breldmg 5330

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 4657	. . . . 5 |- ( x = B -> ( A R x <-> A R B ) )
2	1	spcegv 3294	. . . 4 |- ( B e. D -> ( A R B -> E. x A R x ) )
3	2	imp 445	. . 3 |- ( ( B e. D /\ A R B ) -> E. x A R x )
4		eldmg 5319	. . 3 |- ( A e. C -> ( A e. dom R <-> E. x A R x ) )
5	3, 4	syl5ibr 236	. 2 |- ( A e. C -> ( ( B e. D /\ A R B ) -> A e. dom R ) )
6	5	3impib 1262	1 |- ( ( A e. C /\ B e. D /\ A R B ) -> A e. dom R )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   E.wex 1704   e. wcel 1990   class class class wbr 4653   dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-dm 5124

This theorem is referenced by:  brelrng  5355  releldm  5358  sossfld  5580  brtpos  7361  wfrlem17  7431  tfrlem9a  7482  perpln1  25605  lmdvg  29999  esumcvgsum  30150  fvelimad  39458  climeldmeq  39897  climfv  39923  climxlim2  40072  sge0isum  40644  smflimsuplem6  41031  tz6.12-afv  41253  rlimdmafv  41257