Theorem relun 3267

Description: The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
relun	|- (Rel (A u. B) <-> (Rel A /\ Rel B))

Proof of Theorem relun

Step	Hyp	Ref	Expression
1		unss 2207	. 2 |- ((A (_ (V X. V) /\ B (_ (V X. V)) <-> (A u. B) (_ (V X. V))
2		df-rel 3191	. . 3 |- (Rel A <-> A (_ (V X. V))
3		df-rel 3191	. . 3 |- (Rel B <-> B (_ (V X. V))
4	2, 3	anbi12i 484	. 2 |- ((Rel A /\ Rel B) <-> (A (_ (V X. V) /\ B (_ (V X. V)))
5		df-rel 3191	. 2 |- (Rel (A u. B) <-> (A u. B) (_ (V X. V))
6	1, 4, 5	3bitr4r 184	1 |- (Rel (A u. B) <-> (Rel A /\ Rel B))

Syntax hints:   <-> wb 146   /\ wa 223  Vcvv 1814   u. cun 2048   (_ wss 2050   X. cxp 3174  Rel wrel 3181

This theorem is referenced by:  cnvun 3461  funun 3560

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-in 2054  df-ss 2056  df-rel 3191

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