Theorem reliin 5240

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 4571	. 2 |- ( E. x e. A B C_ ( _V X. _V ) -> |^|_ x e. A B C_ ( _V X. _V ) )
2		df-rel 5121	. . 3 |- ( Rel B <-> B C_ ( _V X. _V ) )
3	2	rexbii 3041	. 2 |- ( E. x e. A Rel B <-> E. x e. A B C_ ( _V X. _V ) )
4		df-rel 5121	. 2 |- ( Rel |^|_ x e. A B <-> |^|_ x e. A B C_ ( _V X. _V ) )
5	1, 3, 4	3imtr4i 281	1 |- ( E. x e. A Rel B -> Rel |^|_ x e. A B )

Syntax hints:   -> wi 4   E.wrex 2913   _Vcvv 3200   C_ wss 3574   |^|_ciin 4521   X. cxp 5112   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-iin 4523  df-rel 5121

This theorem is referenced by:  relint  5242  xpiindi  5257  dibglbN  36455  dihglbcpreN  36589