Theorem resabs1 2592

Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
resabs1	⊢ (B ⊆ C → ((A ↾ C) ↾ B) = (A ↾ B))

Proof of Theorem resabs1

Step	Hyp	Ref	Expression
1		sseqin2 1656	. . . . . 6 ⊢ (B ⊆ C ↔ (C ∩ B) = B)
2		xpeq1 2440	. . . . . 6 ⊢ ((C ∩ B) = B → ((C ∩ B) × V) = (B × V))
3	1, 2	sylbi 174	. . . . 5 ⊢ (B ⊆ C → ((C ∩ B) × V) = (B × V))
4		inxp 2496	. . . . . 6 ⊢ ((C × V) ∩ (B × V)) = ((C ∩ B) × (V ∩ V))
5		inidm 1649	. . . . . . 7 ⊢ (V ∩ V) = V
6		xpeq2 2441	. . . . . . 7 ⊢ ((V ∩ V) = V → ((C ∩ B) × (V ∩ V)) = ((C ∩ B) × V))
7	5, 6	ax-mp 6	. . . . . 6 ⊢ ((C ∩ B) × (V ∩ V)) = ((C ∩ B) × V)
8	4, 7	eqtr2 1120	. . . . 5 ⊢ ((C ∩ B) × V) = ((C × V) ∩ (B × V))
9	3, 8	syl5reqr 1139	. . . 4 ⊢ (B ⊆ C → (B × V) = ((C × V) ∩ (B × V)))
10	9	ineq2d 1645	. . 3 ⊢ (B ⊆ C → (A ∩ (B × V)) = (A ∩ ((C × V) ∩ (B × V))))
11		inass 1650	. . 3 ⊢ ((A ∩ (C × V)) ∩ (B × V)) = (A ∩ ((C × V) ∩ (B × V)))
12	10, 11	syl6reqr 1143	. 2 ⊢ (B ⊆ C → ((A ∩ (C × V)) ∩ (B × V)) = (A ∩ (B × V)))
13		df-res 2430	. . 3 ⊢ ((A ↾ C) ↾ B) = ((A ↾ C) ∩ (B × V))
14		df-res 2430	. . . 4 ⊢ (A ↾ C) = (A ∩ (C × V))
15	14	ineq1i 1641	. . 3 ⊢ ((A ↾ C) ∩ (B × V)) = ((A ∩ (C × V)) ∩ (B × V))
16	13, 15	eqtr 1119	. 2 ⊢ ((A ↾ C) ↾ B) = ((A ∩ (C × V)) ∩ (B × V))
17		df-res 2430	. 2 ⊢ (A ↾ B) = (A ∩ (B × V))
18	12, 16, 17	3eqtr4g 1147	1 ⊢ (B ⊆ C → ((A ↾ C) ↾ B) = (A ↾ B))

Syntax hints:   → wi 2   = wceq 1091  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487   × cxp 2408   ↾ cres 2412

This theorem is referenced by:  fun2ssres 2699  fvres 2840  tfrlem5 2953

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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