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Theorem resdmres 6228
Description: Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.)
Assertion
Ref Expression
resdmres (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)

Proof of Theorem resdmres
StepHypRef Expression
1 in12 4219 . . . 4 (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V)))
2 df-res 5687 . . . . . 6 (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V))
3 resdm2 6227 . . . . . 6 (𝐴 ↾ dom 𝐴) = 𝐴
42, 3eqtr3i 2763 . . . . 5 (𝐴 ∩ (dom 𝐴 × V)) = 𝐴
54ineq2i 4208 . . . 4 ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ 𝐴)
6 incom 4200 . . . 4 ((𝐵 × V) ∩ 𝐴) = (𝐴 ∩ (𝐵 × V))
71, 5, 63eqtri 2765 . . 3 (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (𝐴 ∩ (𝐵 × V))
8 df-res 5687 . . . 4 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴 ∩ (dom (𝐴𝐵) × V))
9 dmres 6001 . . . . . . 7 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
109xpeq1i 5701 . . . . . 6 (dom (𝐴𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V)
11 xpindir 5832 . . . . . 6 ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
1210, 11eqtri 2761 . . . . 5 (dom (𝐴𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
1312ineq2i 4208 . . . 4 (𝐴 ∩ (dom (𝐴𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
148, 13eqtri 2761 . . 3 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
15 df-res 5687 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
167, 14, 153eqtr4i 2771 . 2 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
17 rescnvcnv 6200 . 2 (𝐴𝐵) = (𝐴𝐵)
1816, 17eqtri 2761 1 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3475  cin 3946   × cxp 5673  ccnv 5674  dom cdm 5675  cres 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687
This theorem is referenced by:  resresdm  6229  imadmres  6230  lindfres  21362  imacmp  22883  metreslem  23850  volres  25027  eccnvepres3  37092
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