Theorem reliin 4726

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 3917	. 2 |- ( E. x e. A B C_ ( _V X. _V ) -> |^|_ x e. A B C_ ( _V X. _V ) )
2		df-rel 4611	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
3	2	rexbii 2473	. 2 |- ( E. x e. A Rel B <-> E. x e. A B C_ ( _V X. _V ) )
4		df-rel 4611	. 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 2445   _Vcvv 2726   C_ wss 3116   |^|_ciin 3867   X. cxp 4602   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-iin 3869  df-rel 4611

This theorem is referenced by:  relint  4728  xpiindim  4741