Theorem relss 4751

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 3191	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V)))
2		df-rel 4671	. 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3		df-rel 4671	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
4	1, 2, 3	3imtr4g 205	1 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))

Syntax hints:   → wi 4  Vcvv 2763   ⊆ wss 3157   × cxp 4662  Rel wrel 4669

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-ss 3170  df-rel 4671

This theorem is referenced by:  relin1  4782  relin2  4783  reldif  4784  relres  4975  iss  4993  cnvdif  5077  funss  5278  funssres  5301  fliftcnv  5845  fliftfun  5846  reltpos  6317  tpostpos  6331  swoer  6629  erinxp  6677  ltrel  8105  lerel  8107  txdis1cn  14598  xmeter  14756  lgsquadlem1  15402  lgsquadlem2  15403