Theorem resres 3383

Description: The restriction of a restriction.

Assertion

Ref	Expression
resres	⊢ ((A ↾ B) ↾ C) = (A ↾ (B ∩ C))

Proof of Theorem resres

Step	Hyp	Ref	Expression
1		df-res 3196	. 2 ⊢ ((A ↾ B) ↾ C) = ((A ↾ B) ∩ (C × V))
2		df-res 3196	. . 3 ⊢ (A ↾ B) = (A ∩ (B × V))
3	2	ineq1i 2216	. 2 ⊢ ((A ↾ B) ∩ (C × V)) = ((A ∩ (B × V)) ∩ (C × V))
4		xpindir 3277	. . . 4 ⊢ ((B ∩ C) × V) = ((B × V) ∩ (C × V))
5	4	ineq2i 2217	. . 3 ⊢ (A ∩ ((B ∩ C) × V)) = (A ∩ ((B × V) ∩ (C × V)))
6		df-res 3196	. . 3 ⊢ (A ↾ (B ∩ C)) = (A ∩ ((B ∩ C) × V))
7		inass 2226	. . 3 ⊢ ((A ∩ (B × V)) ∩ (C × V)) = (A ∩ ((B × V) ∩ (C × V)))
8	5, 6, 7	3eqtr4r 1509	. 2 ⊢ ((A ∩ (B × V)) ∩ (C × V)) = (A ↾ (B ∩ C))
9	1, 3, 8	3eqtr 1502	1 ⊢ ((A ↾ B) ↾ C) = (A ↾ (B ∩ C))

Syntax hints:   = wceq 958  Vcvv 1814   ∩ cin 2049   × cxp 3174   ↾ cres 3178

This theorem is referenced by:  rescom 3390  resabs1 3394  curry1 4104

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190  df-rel 3191  df-res 3196

Copyright terms: Public domain