Theorem reliin 4815

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 3993	. 2 |- ( E. x e. A B C_ ( _V X. _V ) -> |^|_ x e. A B C_ ( _V X. _V ) )
2		df-rel 4700	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
3	2	rexbii 2515	. 2 |- ( E. x e. A Rel B <-> E. x e. A B C_ ( _V X. _V ) )
4		df-rel 4700	. 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 2487   _Vcvv 2776   C_ wss 3174   |^|_ciin 3942   X. cxp 4691   Rel wrel 4698

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-in 3180  df-ss 3187  df-iin 3944  df-rel 4700

This theorem is referenced by:  relint  4817  xpiindim  4833