Theorem reluni 4774

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 3962	. . 3 |- U. A = U_ x e. A x
2	1	releqi 4734	. 2 |- ( Rel U. A <-> Rel U_ x e. A x )
3		reliun 4772	. 2 |- ( Rel U_ x e. A x <-> A. x e. A Rel x )
4	2, 3	bitri 184	1 |- ( Rel U. A <-> A. x e. A Rel x )

Syntax hints:   <-> wb 105   A.wral 2468   U.cuni 3831   U_ciun 3908   Rel wrel 4656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2758  df-in 3154  df-ss 3161  df-uni 3832  df-iun 3910  df-rel 4658

This theorem is referenced by:  fununi  5310  tfrlem6  6349