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Theorem resdmres 6188
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 4184 . . . 4 (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V)))
2 df-res 5649 . . . . . 6 (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V))
3 resdm2 6187 . . . . . 6 (𝐴 ↾ dom 𝐴) = 𝐴
42, 3eqtr3i 2763 . . . . 5 (𝐴 ∩ (dom 𝐴 × V)) = 𝐴
54ineq2i 4173 . . . 4 ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ 𝐴)
6 incom 4165 . . . 4 ((𝐵 × V) ∩ 𝐴) = (𝐴 ∩ (𝐵 × V))
71, 5, 63eqtri 2765 . . 3 (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (𝐴 ∩ (𝐵 × V))
8 df-res 5649 . . . 4 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴 ∩ (dom (𝐴𝐵) × V))
9 dmres 5963 . . . . . . 7 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
109xpeq1i 5663 . . . . . 6 (dom (𝐴𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V)
11 xpindir 5794 . . . . . 6 ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
1210, 11eqtri 2761 . . . . 5 (dom (𝐴𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
1312ineq2i 4173 . . . 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 5649 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
167, 14, 153eqtr4i 2771 . 2 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
17 rescnvcnv 6160 . 2 (𝐴𝐵) = (𝐴𝐵)
1816, 17eqtri 2761 1 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3447  cin 3913   × cxp 5635  ccnv 5636  dom cdm 5637  cres 5639
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 5260  ax-nul 5267  ax-pr 5388
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 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648  df-res 5649
This theorem is referenced by:  resresdm  6189  imadmres  6190  lindfres  21252  imacmp  22771  metreslem  23738  volres  24915  eccnvepres3  36796
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