Theorem reliin 4797

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 3979	. 2 |- ( E. x e. A B C_ ( _V X. _V ) -> |^|_ x e. A B C_ ( _V X. _V ) )
2		df-rel 4682	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
3	2	rexbii 2513	. 2 |- ( E. x e. A Rel B <-> E. x e. A B C_ ( _V X. _V ) )
4		df-rel 4682	. 2 |- ( Rel |^|_ x e. A B <-> |^|_ x e. A B C_ ( _V X. _V ) )
5	1, 3, 4	3imtr4i 201	1 |- ( E. x e. A Rel B -> Rel |^|_ x e. A B )

Syntax hints:   -> wi 4   E.wrex 2485   _Vcvv 2772   C_ wss 3166   |^|_ciin 3928   X. cxp 4673   Rel wrel 4680

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-in 3172  df-ss 3179  df-iin 3930  df-rel 4682

This theorem is referenced by:  relint  4799  xpiindim  4815