Theorem relint 4461

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 4459	. 2 |- ( E. x e. A Rel x -> Rel |^|_ x e. A x )
2		intiin 3711	. . 3 |- |^| A = |^|_ x e. A x
3	2	releqi 4423	. 2 |- ( Rel |^| A <-> Rel |^|_ x e. A x )
4	1, 3	sylibr 137	1 |- ( E. x e. A Rel x -> Rel |^| A )

Syntax hints:   -> wi 4   E.wrex 2307   |^|cint 3615   |^|_ciin 3658   Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-int 3616  df-iin 3660  df-rel 4352

This theorem is referenced by: (None)