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Theorem reliin 4487
Description: An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014.)
Assertion
Ref Expression
reliin  |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )

Proof of Theorem reliin
StepHypRef Expression
1 iinss 3736 . 2  |-  ( E. x  e.  A  B  C_  ( _V  X.  _V )  ->  |^|_ x  e.  A  B  C_  ( _V  X.  _V ) )
2 df-rel 4380 . . 3  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
32rexbii 2348 . 2  |-  ( E. x  e.  A  Rel  B  <->  E. x  e.  A  B  C_  ( _V  X.  _V ) )
4 df-rel 4380 . 2  |-  ( Rel  |^|_ x  e.  A  B  <->  |^|_
x  e.  A  B  C_  ( _V  X.  _V ) )
51, 3, 43imtr4i 194 1  |-  ( E. x  e.  A  Rel  B  ->  Rel  |^|_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wrex 2324   _Vcvv 2574    C_ wss 2945   |^|_ciin 3686    X. cxp 4371   Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-iin 3688  df-rel 4380
This theorem is referenced by:  relint  4489  xpiindim  4501
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