Theorem rescom 2588

Description: Commutative law for restriction.

Assertion

Ref	Expression
rescom	|- ((A |` B) |` C) = ((A |` C) |` B)

Proof of Theorem rescom

Step	Hyp	Ref	Expression
1		in23 1652	. . 3 |- ((A i^i (B X. V)) i^i (C X. V)) = ((A i^i (C X. V)) i^i (B X. V))
2		df-res 2430	. . . 4 |- (A |` B) = (A i^i (B X. V))
3	2	ineq1i 1641	. . 3 |- ((A |` B) i^i (C X. V)) = ((A i^i (B X. V)) i^i (C X. V))
4		df-res 2430	. . . 4 |- (A |` C) = (A i^i (C X. V))
5	4	ineq1i 1641	. . 3 |- ((A |` C) i^i (B X. V)) = ((A i^i (C X. V)) i^i (B X. V))
6	1, 3, 5	3eqtr4 1126	. 2 |- ((A |` B) i^i (C X. V)) = ((A |` C) i^i (B X. V))
7		df-res 2430	. 2 |- ((A |` B) |` C) = ((A |` B) i^i (C X. V))
8		df-res 2430	. 2 |- ((A |` C) |` B) = ((A |` C) i^i (B X. V))
9	6, 7, 8	3eqtr4 1126	1 |- ((A |` B) |` C) = ((A |` C) |` B)

Syntax hints:   = wceq 1091  Vcvv 1348   i^i cin 1486   X. cxp 2408   |` cres 2412

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-res 2430

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