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Theorem relss 4525
Description: Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)
Assertion
Ref Expression
relss (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))

Proof of Theorem relss
StepHypRef Expression
1 sstr2 3032 . 2 (𝐴𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V)))
2 df-rel 4445 . 2 (Rel 𝐵𝐵 ⊆ (V × V))
3 df-rel 4445 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
41, 2, 33imtr4g 203 1 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  Vcvv 2619  wss 2999   × cxp 4436  Rel wrel 4443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012  df-rel 4445
This theorem is referenced by:  relin1  4555  relin2  4556  reldif  4557  relres  4741  iss  4758  cnvdif  4838  funss  5034  funssres  5056  fliftcnv  5574  fliftfun  5575  reltpos  6015  tpostpos  6029  swoer  6320  erinxp  6366  ltrel  7548  lerel  7550
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