Theorem relss 4805

Description: Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)

Assertion

Ref	Expression
relss	⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))

Proof of Theorem relss

Step	Hyp	Ref	Expression
1		sstr2 3231	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V)))
2		df-rel 4725	. 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3		df-rel 4725	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
4	1, 2, 3	3imtr4g 205	1 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))

Syntax hints:   → wi 4  Vcvv 2799   ⊆ wss 3197   × cxp 4716  Rel wrel 4723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-in 3203  df-ss 3210  df-rel 4725

This theorem is referenced by:  relin1  4836  relin2  4837  reldif  4838  relres  5032  iss  5050  cnvdif  5134  funss  5336  funssres  5359  fliftcnv  5918  fliftfun  5919  reltpos  6394  tpostpos  6408  swoer  6706  erinxp  6754  ltrel  8204  lerel  8206  txdis1cn  14946  xmeter  15104  lgsquadlem1  15750  lgsquadlem2  15751