Theorem relres 3371

Description: A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
relres	|- Rel (A |` B)

Proof of Theorem relres

Step	Hyp	Ref	Expression
1		df-res 3180	. . 3 |- (A |` B) = (A i^i (B X. V))
2		inss2 2221	. . . 4 |- (A i^i (B X. V)) (_ (B X. V)
3		xpss 3220	. . . 4 |- (B X. V) (_ (V X. V)
4	2, 3	sstri 2063	. . 3 |- (A i^i (B X. V)) (_ (V X. V)
5	1, 4	eqsstr 2081	. 2 |- (A |` B) (_ (V X. V)
6		df-rel 3175	. 2 |- (Rel (A |` B) <-> (A |` B) (_ (V X. V))
7	5, 6	mpbir 190	1 |- Rel (A |` B)

Syntax hints:  Vcvv 1802   i^i cin 2036   (_ wss 2037   X. cxp 3158   |` cres 3162  Rel wrel 3165

This theorem is referenced by:  resiexg 3380  iss 3381  asymref 3423  asymrefOLD 3425  cnvcnvres 3480  resco 3486  cores2 3493  unidmrnOLD 3502  funssres 3538  resfunexg 3565  fnresdisj 3583  fcnvres 3633

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-opab 2657  df-xp 3174  df-rel 3175  df-res 3180

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