Theorem reliin 4661

Description: An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)

Assertion

Ref	Expression
reliin	|- ( E. x e. A Rel B -> Rel |^|_ x e. A B )

Proof of Theorem reliin

Step	Hyp	Ref	Expression
1		iinss 3864	. 2 |- ( E. x e. A B C_ ( _V X. _V ) -> |^|_ x e. A B C_ ( _V X. _V ) )
2		df-rel 4546	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
3	2	rexbii 2442	. 2 |- ( E. x e. A Rel B <-> E. x e. A B C_ ( _V X. _V ) )
4		df-rel 4546	. 2 |- ( Rel |^|_ x e. A B <-> |^|_ x e. A B C_ ( _V X. _V ) )
5	1, 3, 4	3imtr4i 200	1 |- ( E. x e. A Rel B -> Rel |^|_ x e. A B )

Syntax hints:   -> wi 4   E.wrex 2417   _Vcvv 2686   C_ wss 3071   |^|_ciin 3814   X. cxp 4537   Rel wrel 4544

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-iin 3816  df-rel 4546

This theorem is referenced by:  relint  4663  xpiindim  4676