Theorem resabs2 2593

Description: Absorption law for restriction.

Assertion

Ref	Expression
resabs2	|- (B (_ C -> ((A |` B) |` C) = (A |` B))

Proof of Theorem resabs2

Step	Hyp	Ref	Expression
1		ssid 1519	. . . . 5 |- V (_ V
2		ssxp 2487	. . . . . 6 |- ((B (_ C /\ V (_ V) -> (B X. V) (_ (C X. V))
3		dfss 1493	. . . . . 6 |- ((B X. V) (_ (C X. V) <-> (B X. V) = ((B X. V) i^i (C X. V)))
4	2, 3	sylib 173	. . . . 5 |- ((B (_ C /\ V (_ V) -> (B X. V) = ((B X. V) i^i (C X. V)))
5	1, 4	mpan2 519	. . . 4 |- (B (_ C -> (B X. V) = ((B X. V) i^i (C X. V)))
6	5	ineq2d 1645	. . 3 |- (B (_ C -> (A i^i (B X. V)) = (A i^i ((B X. V) i^i (C X. V))))
7		inass 1650	. . 3 |- ((A i^i (B X. V)) i^i (C X. V)) = (A i^i ((B X. V) i^i (C X. V)))
8	6, 7	syl6reqr 1143	. 2 |- (B (_ C -> ((A i^i (B X. V)) i^i (C X. V)) = (A i^i (B X. V)))
9		df-res 2430	. . 3 |- ((A |` B) |` C) = ((A |` B) i^i (C X. V))
10		df-res 2430	. . . 4 |- (A |` B) = (A i^i (B X. V))
11	10	ineq1i 1641	. . 3 |- ((A |` B) i^i (C X. V)) = ((A i^i (B X. V)) i^i (C X. V))
12	9, 11	eqtr 1119	. 2 |- ((A |` B) |` C) = ((A i^i (B X. V)) i^i (C X. V))
13	8, 12, 10	3eqtr4g 1147	1 |- (B (_ C -> ((A |` B) |` C) = (A |` B))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091  Vcvv 1348   i^i cin 1486   (_ wss 1487   X. cxp 2408   |` cres 2412

This theorem is referenced by:  residm 2594

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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