Theorem relres 5091

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 4786	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss2 3452	. . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
3	1, 2	eqsstri 3280	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V)
4		relxp 4884	. 2 ⊢ Rel (𝐵 × V)
5		relss 4862	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵)))
6	3, 4, 5	mp2 16	1 ⊢ Rel (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:  Vcvv 2821   ∩ cin 3219   ⊆ wss 3220   × cxp 4772   ↾ cres 4776  Rel wrel 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-opab 4193  df-xp 4780  df-rel 4781  df-res 4786

This theorem is used by:  elres  5099  resiexg  5108  iss  5109  dfres2  5115  restidsing  5119  issref  5170  asymref  5173  poirr2  5180  cnvcnvres  5251  resco  5292  ressn  5328  funssres  5420  fnresdisj  5493  fnres  5500  fcnvres  5575  nfunsn  5733  fsnunfv  5916  resfunexgALT  6337  setsresg  13390