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Theorem relint 4489
 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 (∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem relint
StepHypRef Expression
1 reliin 4487 . 2 (∃𝑥𝐴 Rel 𝑥 → Rel 𝑥𝐴 𝑥)
2 intiin 3739 . . 3 𝐴 = 𝑥𝐴 𝑥
32releqi 4451 . 2 (Rel 𝐴 ↔ Rel 𝑥𝐴 𝑥)
41, 3sylibr 141 1 (∃𝑥𝐴 Rel 𝑥 → Rel 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∃wrex 2324  ∩ cint 3643  ∩ ciin 3686  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-int 3644  df-iin 3688  df-rel 4380 This theorem is referenced by: (None)
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