Theorem brelg 3222

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 3216	. . 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 2065	. . 3 |- (<.A, B>. e. R -> <.A, B>. e. (C X. D))
4	1, 3	syl5bi 208	. 2 |- (B e. S -> (<.A, B>. e. R -> (A e. C /\ B e. D)))
5		df-br 2620	. 2 |- (ARB <-> <.A, B>. e. R)
6	4, 5	syl5ib 206	1 |- (B e. S -> (ARB -> (A e. C /\ B e. D)))

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958   (_ wss 2047  <.cop 2411   class class class wbr 2619   X. cxp 3168

This theorem is referenced by:  brel 3223  suplem2pr 5162

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184

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