Theorem reluni 4478

Description: The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.)

Assertion

Ref	Expression
reluni	|- ( Rel U. A <-> A. x e. A Rel x )

Distinct variable group:   x, A

Proof of Theorem reluni

Step	Hyp	Ref	Expression
1		uniiun 3731	. . 3 |- U. A = U_ x e. A x
2	1	releqi 4441	. 2 |- ( Rel U. A <-> Rel U_ x e. A x )
3		reliun 4476	. 2 |- ( Rel U_ x e. A x <-> A. x e. A Rel x )
4	2, 3	bitri 182	1 |- ( Rel U. A <-> A. x e. A Rel x )

Syntax hints:   <-> wb 103   A.wral 2348   U.cuni 3601   U_ciun 3678   Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-uni 3602  df-iun 3680  df-rel 4370

This theorem is referenced by:  fununi  4987  tfrlem6  5955