Theorem relint 5242

Description: The intersection of a class is a relation if at least one member is a relation. (Contributed by NM, 8-Mar-2014.)

Assertion

Ref	Expression
relint	|- ( E. x e. A Rel x -> Rel |^| A )

Distinct variable group:   x, A

Proof of Theorem relint

Step	Hyp	Ref	Expression
1		reliin 5240	. 2 |- ( E. x e. A Rel x -> Rel |^|_ x e. A x )
2		intiin 4574	. . 3 |- |^| A = |^|_ x e. A x
3	2	releqi 5202	. 2 |- ( Rel |^| A <-> Rel |^|_ x e. A x )
4	1, 3	sylibr 224	1 |- ( E. x e. A Rel x -> Rel |^| A )

Syntax hints:   -> wi 4   E.wrex 2913   |^|cint 4475   |^|_ciin 4521   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-int 4476  df-iin 4523  df-rel 5121

This theorem is referenced by:  clrellem  37929