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Theorem resdmres 6082
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 4194 . . . 4 (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V)))
2 df-res 5560 . . . . . 6 (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V))
3 resdm2 6081 . . . . . 6 (𝐴 ↾ dom 𝐴) = 𝐴
42, 3eqtr3i 2843 . . . . 5 (𝐴 ∩ (dom 𝐴 × V)) = 𝐴
54ineq2i 4183 . . . 4 ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ 𝐴)
6 incom 4175 . . . 4 ((𝐵 × V) ∩ 𝐴) = (𝐴 ∩ (𝐵 × V))
71, 5, 63eqtri 2845 . . 3 (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (𝐴 ∩ (𝐵 × V))
8 df-res 5560 . . . 4 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴 ∩ (dom (𝐴𝐵) × V))
9 dmres 5868 . . . . . . 7 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
109xpeq1i 5574 . . . . . 6 (dom (𝐴𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V)
11 xpindir 5698 . . . . . 6 ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
1210, 11eqtri 2841 . . . . 5 (dom (𝐴𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
1312ineq2i 4183 . . . 4 (𝐴 ∩ (dom (𝐴𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
148, 13eqtri 2841 . . 3 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
15 df-res 5560 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
167, 14, 153eqtr4i 2851 . 2 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
17 rescnvcnv 6054 . 2 (𝐴𝐵) = (𝐴𝐵)
1816, 17eqtri 2841 1 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  Vcvv 3492  cin 3932   × cxp 5546  ccnv 5547  dom cdm 5548  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560
This theorem is referenced by:  resresdm  6083  imadmres  6084  lindfres  20895  imacmp  21933  metreslem  22899  volres  24056  eccnvepres3  35423
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