Theorem relres 4986

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 4686	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss2 3393	. . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
3	1, 2	eqsstri 3224	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V)
4		relxp 4783	. 2 ⊢ Rel (𝐵 × V)
5		relss 4761	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵)))
6	3, 4, 5	mp2 16	1 ⊢ Rel (𝐴 ↾ 𝐵)

Syntax hints:  Vcvv 2771   ∩ cin 3164   ⊆ wss 3165   × cxp 4672   ↾ cres 4676  Rel wrel 4679

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-opab 4105  df-xp 4680  df-rel 4681  df-res 4686

This theorem is referenced by:  elres  4994  resiexg  5003  iss  5004  dfres2  5010  restidsing  5014  issref  5064  asymref  5067  poirr2  5074  cnvcnvres  5145  resco  5186  ressn  5222  funssres  5312  fnresdisj  5385  fnres  5391  fcnvres  5458  nfunsn  5610  fsnunfv  5784  resfunexgALT  6192  setsresg  12841