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Theorem relint 4404
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 (x A Rel x → Rel A)
Distinct variable group:   x,A

Proof of Theorem relint
StepHypRef Expression
1 reliin 4402 . 2 (x A Rel x → Rel x A x)
2 intiin 3702 . . 3 A = x A x
32releqi 4366 . 2 (Rel A ↔ Rel x A x)
41, 3sylibr 137 1 (x A Rel x → Rel A)
Colors of variables: wff set class
Syntax hints:  wi 4  wrex 2301   cint 3606   ciin 3649  Rel wrel 4293
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-int 3607  df-iin 3651  df-rel 4295
This theorem is referenced by: (None)
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