Theorem relres 2591

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 2430	. . 3 |- (A |` B) = (A i^i (B X. V))
2		inss2 1658	. . . 4 |- (A i^i (B X. V)) (_ (B X. V)
3		xpss 2465	. . . 4 |- (B X. V) (_ (V X. V)
4	2, 3	sstri 1512	. . 3 |- (A i^i (B X. V)) (_ (V X. V)
5	1, 4	eqsstr 1530	. 2 |- (A |` B) (_ (V X. V)
6		df-rel 2425	. 2 |- (Rel (A |` B) <-> (A |` B) (_ (V X. V))
7	5, 6	mpbir 165	1 |- Rel (A |` B)

Syntax hints:  Vcvv 1348   i^i cin 1486   (_ wss 1487   X. cxp 2408   |` cres 2412  Rel wrel 2415

This theorem is referenced by:  iss 2599  funssres 2698  resfunexg 2717  fnresdisj 2732  fcnvres 2768

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-res 2430

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