Theorem reluni 4460

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 3710	. . 3 |- U. A = U_ x e. A x
2	1	releqi 4423	. 2 |- ( Rel U. A <-> Rel U_ x e. A x )
3		reliun 4458	. 2 |- ( Rel U_ x e. A x <-> A. x e. A Rel x )
4	2, 3	bitri 173	1 |- ( Rel U. A <-> A. x e. A Rel x )

Syntax hints:   <-> wb 98   A.wral 2306   U.cuni 3580   U_ciun 3657   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-iun 3659  df-rel 4352

This theorem is referenced by:  fununi  4967  tfrlem6  5932