Theorem relres 4987

Description: A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
relres	⊢ Rel (𝐴 ↾ 𝐵)

Proof of Theorem relres

Step	Hyp	Ref	Expression
1		df-res 4687	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss2 3394	. . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
3	1, 2	eqsstri 3225	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V)
4		relxp 4784	. 2 ⊢ Rel (𝐵 × V)
5		relss 4762	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵)))
6	3, 4, 5	mp2 16	1 ⊢ Rel (𝐴 ↾ 𝐵)

Syntax hints:  Vcvv 2772   ∩ cin 3165   ⊆ wss 3166   × cxp 4673   ↾ cres 4677  Rel wrel 4680

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-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-in 3172  df-ss 3179  df-opab 4106  df-xp 4681  df-rel 4682  df-res 4687

This theorem is referenced by:  elres  4995  resiexg  5004  iss  5005  dfres2  5011  restidsing  5015  issref  5065  asymref  5068  poirr2  5075  cnvcnvres  5146  resco  5187  ressn  5223  funssres  5313  fnresdisj  5386  fnres  5392  fcnvres  5459  nfunsn  5611  fsnunfv  5785  resfunexgALT  6193  setsresg  12870