Theorem brelg 2458

Description: Two things in a binary relation belong to the relation's domain.

Hypothesis

Ref	Expression
brelg.1	|- R (_ (C X. D)

Assertion

Ref	Expression
brelg	|- (B e. S -> (ARB -> (A e. C /\ B e. D)))

Proof of Theorem brelg

Step	Hyp	Ref	Expression
1		opelxpg 2454	. . 3 |- (B e. S -> (<.A, B>. e. (C X. D) <-> (A e. C /\ B e. D)))
2		brelg.1	. . . 4 |- R (_ (C X. D)
3	2	sseli 1504	. . 3 |- (<.A, B>. e. R -> <.A, B>. e. (C X. D))
4	1, 3	syl5bi 183	. 2 |- (B e. S -> (<.A, B>. e. R -> (A e. C /\ B e. D)))
5		df-br 2063	. 2 |- (ARB <-> <.A, B>. e. R)
6	4, 5	syl5ib 181	1 |- (B e. S -> (ARB -> (A e. C /\ B e. D)))

Syntax hints:   -> wi 2   /\ wa 196   e. wcel 1092   (_ wss 1487  <.cop 1810   class class class wbr 2054   X. cxp 2408

This theorem is referenced by:  brel 2459  suplem2pr 3956

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424

metamath.org