Theorem reluni 4786

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 3970	. . 3 |- U. A = U_ x e. A x
2	1	releqi 4746	. 2 |- ( Rel U. A <-> Rel U_ x e. A x )
3		reliun 4784	. 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 2475   U.cuni 3839   U_ciun 3916   Rel wrel 4668

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-iun 3918  df-rel 4670

This theorem is referenced by:  fununi  5326  tfrlem6  6374