Theorem relint 4735

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 4733	. 2 |- ( E. x e. A Rel x -> Rel |^|_ x e. A x )
2		intiin 3927	. . 3 |- |^| A = |^|_ x e. A x
3	2	releqi 4694	. 2 |- ( Rel |^| A <-> Rel |^|_ x e. A x )
4	1, 3	sylibr 133	1 |- ( E. x e. A Rel x -> Rel |^| A )

Syntax hints:   -> wi 4   E.wrex 2449   |^|cint 3831   |^|_ciin 3874   Rel wrel 4616

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-int 3832  df-iin 3876  df-rel 4618

This theorem is referenced by: (None)