Theorem resdmres 6235

Description: Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)

Assertion

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

Proof of Theorem resdmres

Step	Hyp	Ref	Expression
1		in12 4181	. . . 4 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V)))
2		df-res 5675	. . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V))
3		resdm2 6234	. . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴
4	2, 3	eqtr3i 2790	. . . . 5 ⊢ (𝐴 ∩ (dom 𝐴 × V)) = ◡◡𝐴
5	4	ineq2i 4170	. . . 4 ⊢ ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ ◡◡𝐴)
6		incom 4162	. . . 4 ⊢ ((𝐵 × V) ∩ ◡◡𝐴) = (◡◡𝐴 ∩ (𝐵 × V))
7	1, 5, 6	3eqtri 2792	. . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (◡◡𝐴 ∩ (𝐵 × V))
8		df-res 5675	. . . 4 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V))
9		dmres 6013	. . . . . . 7 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
10	9	xpeq1i 5689	. . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V)
11		xpindir 5822	. . . . . 6 ⊢ ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
12	10, 11	eqtri 2788	. . . . 5 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
13	12	ineq2i 4170	. . . 4 ⊢ (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
14	8, 13	eqtri 2788	. . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
15		df-res 5675	. . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (◡◡𝐴 ∩ (𝐵 × V))
16	7, 14, 15	3eqtr4i 2798	. 2 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (◡◡𝐴 ↾ 𝐵)
17		rescnvcnv 6207	. 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵)
18	16, 17	eqtri 2788	1 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457   ∩ cin 3905   × cxp 5661  ^◡ccnv 5662  dom cdm 5663   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675

This theorem is used by:  resresdm  6236  imadmres  6237  lindfres  22023  imacmp  23604  metreslem  24570  volres  25738  eccnvepres3  38999  isubgruhgr  48691