Theorem relres 4937

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 4640	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss2 3358	. . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
3	1, 2	eqsstri 3189	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V)
4		relxp 4737	. 2 ⊢ Rel (𝐵 × V)
5		relss 4715	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵)))
6	3, 4, 5	mp2 16	1 ⊢ Rel (𝐴 ↾ 𝐵)

Syntax hints:  Vcvv 2739   ∩ cin 3130   ⊆ wss 3131   × cxp 4626   ↾ cres 4630  Rel wrel 4633

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-opab 4067  df-xp 4634  df-rel 4635  df-res 4640

This theorem is referenced by:  elres  4945  resiexg  4954  iss  4955  dfres2  4961  restidsing  4965  issref  5013  asymref  5016  poirr2  5023  cnvcnvres  5094  resco  5135  ressn  5171  funssres  5260  fnresdisj  5328  fnres  5334  fcnvres  5401  nfunsn  5551  fsnunfv  5719  resfunexgALT  6111  setsresg  12502