Theorem brelg 29421

Description: Two things in a binary relation belong to the relation's domain. (Contributed by Thierry Arnoux, 29-Aug-2017.)

Assertion

Ref	Expression
brelg	|- ( ( R C_ ( C X. D ) /\ A R B ) -> ( A e. C /\ B e. D ) )

Proof of Theorem brelg

Step	Hyp	Ref	Expression
1		id 22	. . . 4 |- ( R C_ ( C X. D ) -> R C_ ( C X. D ) )
2	1	ssbrd 4696	. . 3 |- ( R C_ ( C X. D ) -> ( A R B -> A ( C X. D ) B ) )
3	2	imp 445	. 2 |- ( ( R C_ ( C X. D ) /\ A R B ) -> A ( C X. D ) B )
4		brxp 5147	. 2 |- ( A ( C X. D ) B <-> ( A e. C /\ B e. D ) )
5	3, 4	sylib 208	1 |- ( ( R C_ ( C X. D ) /\ A R B ) -> ( A e. C /\ B e. D ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   class class class wbr 4653   X. cxp 5112

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-ral 2917  df-rex 2918  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-opab 4713  df-xp 5120

This theorem is referenced by:  fpwrelmap  29508