Theorem relss 4780

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 3208	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (V × V) → 𝐴 ⊆ (V × V)))
2		df-rel 4700	. 2 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V))
3		df-rel 4700	. 2 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V))
4	1, 2, 3	3imtr4g 205	1 ⊢ (𝐴 ⊆ 𝐵 → (Rel 𝐵 → Rel 𝐴))

Syntax hints:   → wi 4  Vcvv 2776   ⊆ wss 3174   × cxp 4691  Rel wrel 4698

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-in 3180  df-ss 3187  df-rel 4700

This theorem is referenced by:  relin1  4811  relin2  4812  reldif  4813  relres  5006  iss  5024  cnvdif  5108  funss  5309  funssres  5332  fliftcnv  5887  fliftfun  5888  reltpos  6359  tpostpos  6373  swoer  6671  erinxp  6719  ltrel  8169  lerel  8171  txdis1cn  14865  xmeter  15023  lgsquadlem1  15669  lgsquadlem2  15670