Theorem resres 3377

Description: The restriction of a restriction.

Assertion

Ref	Expression
resres	|- ((A |` B) |` C) = (A |` (B i^i C))

Proof of Theorem resres

Step	Hyp	Ref	Expression
1		df-res 3190	. 2 |- ((A |` B) |` C) = ((A |` B) i^i (C X. V))
2		df-res 3190	. . 3 |- (A |` B) = (A i^i (B X. V))
3	2	ineq1i 2213	. 2 |- ((A |` B) i^i (C X. V)) = ((A i^i (B X. V)) i^i (C X. V))
4		xpindir 3271	. . . 4 |- ((B i^i C) X. V) = ((B X. V) i^i (C X. V))
5	4	ineq2i 2214	. . 3 |- (A i^i ((B i^i C) X. V)) = (A i^i ((B X. V) i^i (C X. V)))
6		df-res 3190	. . 3 |- (A |` (B i^i C)) = (A i^i ((B i^i C) X. V))
7		inass 2223	. . 3 |- ((A i^i (B X. V)) i^i (C X. V)) = (A i^i ((B X. V) i^i (C X. V)))
8	5, 6, 7	3eqtr4r 1506	. 2 |- ((A i^i (B X. V)) i^i (C X. V)) = (A |` (B i^i C))
9	1, 3, 8	3eqtr 1499	1 |- ((A |` B) |` C) = (A |` (B i^i C))

Syntax hints:   = wceq 956  Vcvv 1811   i^i cin 2046   X. cxp 3168   |` cres 3172

This theorem is referenced by:  rescom 3384  resabs1 3388  curry1 4098

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-opab 2667  df-xp 3184  df-rel 3185  df-res 3190

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