Theorem relint 4787

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 4785	. 2 |- ( E. x e. A Rel x -> Rel |^|_ x e. A x )
2		intiin 3971	. . 3 |- |^| A = |^|_ x e. A x
3	2	releqi 4746	. 2 |- ( Rel |^| A <-> Rel |^|_ x e. A x )
4	1, 3	sylibr 134	1 |- ( E. x e. A Rel x -> Rel |^| A )

Syntax hints:   -> wi 4   E.wrex 2476   |^|cint 3874   |^|_ciin 3917   Rel wrel 4668

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-in 3163  df-ss 3170  df-int 3875  df-iin 3919  df-rel 4670

This theorem is referenced by: (None)