Theorem relint 4812

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 4810	. 2 |- ( E. x e. A Rel x -> Rel |^|_ x e. A x )
2		intiin 3991	. . 3 |- |^| A = |^|_ x e. A x
3	2	releqi 4771	. 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 2486   |^|cint 3894   |^|_ciin 3937   Rel wrel 4693

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 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-in 3176  df-ss 3183  df-int 3895  df-iin 3939  df-rel 4695

This theorem is referenced by: (None)