Theorem relss 4370

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 2946	. 2 ⊢ (A ⊆ B → (B ⊆ (V × V) → A ⊆ (V × V)))
2		df-rel 4295	. 2 ⊢ (Rel B ↔ B ⊆ (V × V))
3		df-rel 4295	. 2 ⊢ (Rel A ↔ A ⊆ (V × V))
4	1, 2, 3	3imtr4g 194	1 ⊢ (A ⊆ B → (Rel B → Rel A))

Syntax hints:   → wi 4  Vcvv 2551   ⊆ wss 2911   × cxp 4286  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925  df-rel 4295

This theorem is referenced by:  relin1  4398  relin2  4399  reldif  4400  relres  4582  iss  4597  cnvdif  4673  funss  4863  funssres  4885  fliftcnv  5378  fliftfun  5379  reltpos  5806  tpostpos  5820  swoer  6070  erinxp  6116  ltrel  6858  lerel  6860