Theorem relss 4842

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 3249	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V)))
2		df-rel 4761	. 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3		df-rel 4761	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
4	1, 2, 3	3imtr4g 205	1 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))

Syntax hints:   → wi 4  Vcvv 2815   ⊆ wss 3214   × cxp 4752  Rel wrel 4759

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-in 3220  df-ss 3227  df-rel 4761

This theorem is referenced by:  relin1  4875  relin2  4876  reldif  4877  relres  5071  iss  5089  cnvdif  5174  funss  5376  funssres  5400  fliftcnv  5974  fliftfun  5975  reltpos  6494  tpostpos  6508  swoer  6808  erinxp  6856  ltrel  8351  lerel  8353  txdis1cn  15269  xmeter  15427  lgsquadlem1  16076  lgsquadlem2  16077