Theorem brrelex 3207

Description: A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.)

Assertion

Ref	Expression
brrelex	|- ((Rel R /\ ARB) -> A e. V)

Proof of Theorem brrelex

Step	Hyp	Ref	Expression
1		df-rel 3185	. . 3 |- (Rel R <-> R (_ (V X. V))
2		ssel 2063	. . . . 5 |- (R (_ (V X. V) -> (<.A, B>. e. R -> <.A, B>. e. (V X. V)))
3		df-br 2620	. . . . 5 |- (ARB <-> <.A, B>. e. R)
4	2, 3	syl5ib 206	. . . 4 |- (R (_ (V X. V) -> (ARB -> <.A, B>. e. (V X. V)))
5		opelxp1 3205	. . . 4 |- (<.A, B>. e. (V X. V) -> A e. V)
6	4, 5	syl6 22	. . 3 |- (R (_ (V X. V) -> (ARB -> A e. V))
7	1, 6	sylbi 199	. 2 |- (Rel R -> (ARB -> A e. V))
8	7	imp 350	1 |- ((Rel R /\ ARB) -> A e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 958  Vcvv 1811   (_ wss 2047  <.cop 2411   class class class wbr 2619   X. cxp 3168  Rel wrel 3175

This theorem is referenced by:  brrelexi 3208  releldm 3346  relelrng 3347  relimasn 3425  fnbr 3590  funbrfv 3750  lmbr 7928  psasym 8651  pstr 8652  spwpr3OLD 8662  spwpr4OLD 8663  spwpr4aOLD 8664

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  df-rel 3185

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