Theorem reluni 4727

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 3919	. . 3 |- U. A = U_ x e. A x
2	1	releqi 4687	. 2 |- ( Rel U. A <-> Rel U_ x e. A x )
3		reliun 4725	. 2 |- ( Rel U_ x e. A x <-> A. x e. A Rel x )
4	2, 3	bitri 183	1 |- ( Rel U. A <-> A. x e. A Rel x )

Syntax hints:   <-> wb 104   A.wral 2444   U.cuni 3789   U_ciun 3866   Rel wrel 4609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-in 3122  df-ss 3129  df-uni 3790  df-iun 3868  df-rel 4611

This theorem is referenced by:  fununi  5256  tfrlem6  6284