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Theorem relss 4454
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 2979 . 2 (𝐴𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V)))
2 df-rel 4379 . 2 (Rel 𝐵𝐵 ⊆ (V × V))
3 df-rel 4379 . 2 (Rel 𝐴𝐴 ⊆ (V × V))
41, 2, 33imtr4g 198 1 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  Vcvv 2574  wss 2944   × cxp 4370  Rel wrel 4377
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-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  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-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2951  df-ss 2958  df-rel 4379
This theorem is referenced by:  relin1  4482  relin2  4483  reldif  4484  relres  4666  iss  4681  cnvdif  4757  funss  4947  funssres  4969  fliftcnv  5462  fliftfun  5463  reltpos  5895  tpostpos  5909  swoer  6164  erinxp  6210  ltrel  7139  lerel  7141
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