Theorem resdmres 3503

Description: Restriction to the domain of a restriction.

Assertion

Ref	Expression
resdmres	⊢ (A ↾ dom ( A ↾ B)) = (A ↾ B)

Proof of Theorem resdmres

Step	Hyp	Ref	Expression
1		in12 2227	. . . 4 ⊢ (A ∩ ((B × V) ∩ (dom A × V))) = ((B × V) ∩ (A ∩ (dom A × V)))
2		df-res 3196	. . . . . 6 ⊢ (A ↾ dom A) = (A ∩ (dom A × V))
3		resdm2 3502	. . . . . 6 ⊢ (A ↾ dom A) = ◡◡A
4	2, 3	eqtr3 1500	. . . . 5 ⊢ (A ∩ (dom A × V)) = ◡◡A
5	4	ineq2i 2217	. . . 4 ⊢ ((B × V) ∩ (A ∩ (dom A × V))) = ((B × V) ∩ ◡◡A)
6		incom 2211	. . . 4 ⊢ ((B × V) ∩ ◡◡A) = (◡◡A ∩ (B × V))
7	1, 5, 6	3eqtr 1502	. . 3 ⊢ (A ∩ ((B × V) ∩ (dom A × V))) = (◡◡A ∩ (B × V))
8		df-res 3196	. . . 4 ⊢ (A ↾ dom ( A ↾ B)) = (A ∩ (dom ( A ↾ B) × V))
9		dmres 3386	. . . . . . 7 ⊢ dom ( A ↾ B) = (B ∩ dom A)
10		xpeq1 3206	. . . . . . 7 ⊢ (dom ( A ↾ B) = (B ∩ dom A) → (dom ( A ↾ B) × V) = ((B ∩ dom A) × V))
11	9, 10	ax-mp 7	. . . . . 6 ⊢ (dom ( A ↾ B) × V) = ((B ∩ dom A) × V)
12		xpindir 3277	. . . . . 6 ⊢ ((B ∩ dom A) × V) = ((B × V) ∩ (dom A × V))
13	11, 12	eqtr 1498	. . . . 5 ⊢ (dom ( A ↾ B) × V) = ((B × V) ∩ (dom A × V))
14	13	ineq2i 2217	. . . 4 ⊢ (A ∩ (dom ( A ↾ B) × V)) = (A ∩ ((B × V) ∩ (dom A × V)))
15	8, 14	eqtr 1498	. . 3 ⊢ (A ↾ dom ( A ↾ B)) = (A ∩ ((B × V) ∩ (dom A × V)))
16		df-res 3196	. . 3 ⊢ (◡◡A ↾ B) = (◡◡A ∩ (B × V))
17	7, 15, 16	3eqtr4 1508	. 2 ⊢ (A ↾ dom ( A ↾ B)) = (◡◡A ↾ B)
18		rescnvcnv 3499	. 2 ⊢ (◡◡A ↾ B) = (A ↾ B)
19	17, 18	eqtr 1498	1 ⊢ (A ↾ dom ( A ↾ B)) = (A ↾ B)

Syntax hints:   = wceq 958  Vcvv 1814   ∩ cin 2049   × cxp 3174  ^◡ccnv 3175  dom cdm 3176   ↾ cres 3178

This theorem is referenced by:  imadmres 3504  metres 7820

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-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196

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