Theorem relres 4242

Description: A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (The proof was 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 4006	. . 3 |- (A |` B) = (A i^i (B X. _V))
2		inss2 2813	. . 3 |- (A i^i (B X. _V)) C_ (B X. _V)
3	1, 2	eqsstri 2647	. 2 |- (A |` B) C_ (B X. _V)
4		relxp 4088	. 2 |- Rel (B X. _V)
5		relss 4074	. 2 |- ((A |` B) C_ (B X. _V) -> (Rel (B X. _V) -> Rel (A |` B)))
6	3, 4, 5	mp2 54	1 |- Rel (A |` B)

Syntax hints:  _Vcvv 2292   i^i cin 2592   C_ wss 2593   X. cxp 3984   |` cres 3988  Rel wrel 3991

This theorem is referenced by:  resiexg 4253  iss 4254  issOLD 4255  asymref 4308  asymrefOLD 4309  cnvcnvres 4387  resco 4402  cores2 4410  funssres 4460  resfunexg 4500  fnresdisj 4523  fcnvres 4589  nfunsn 4706  nfunsnOLD 4707  dffv2 4734  elres 13824  ref4w 14370  restidsing 14391  dispos 14632  filnetlem4 15643

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-opab 3396  df-xp 4000  df-rel 4001  df-res 4006

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