Theorem relres 4953

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 4656	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss2 3371	. . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
3	1, 2	eqsstri 3202	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V)
4		relxp 4753	. 2 ⊢ Rel (𝐵 × V)
5		relss 4731	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵)))
6	3, 4, 5	mp2 16	1 ⊢ Rel (𝐴 ↾ 𝐵)

Syntax hints:  Vcvv 2752   ∩ cin 3143   ⊆ wss 3144   × cxp 4642   ↾ cres 4646  Rel wrel 4649

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-opab 4080  df-xp 4650  df-rel 4651  df-res 4656

This theorem is referenced by:  elres  4961  resiexg  4970  iss  4971  dfres2  4977  restidsing  4981  issref  5029  asymref  5032  poirr2  5039  cnvcnvres  5110  resco  5151  ressn  5187  funssres  5277  fnresdisj  5345  fnres  5351  fcnvres  5418  nfunsn  5568  fsnunfv  5737  resfunexgALT  6132  setsresg  12549