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Theorem resdmres 4839
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 3175 . . . 4 (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V)))
2 df-res 4384 . . . . . 6 (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V))
3 resdm2 4838 . . . . . 6 (𝐴 ↾ dom 𝐴) = 𝐴
42, 3eqtr3i 2078 . . . . 5 (𝐴 ∩ (dom 𝐴 × V)) = 𝐴
54ineq2i 3162 . . . 4 ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ 𝐴)
6 incom 3156 . . . 4 ((𝐵 × V) ∩ 𝐴) = (𝐴 ∩ (𝐵 × V))
71, 5, 63eqtri 2080 . . 3 (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (𝐴 ∩ (𝐵 × V))
8 df-res 4384 . . . 4 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴 ∩ (dom (𝐴𝐵) × V))
9 dmres 4659 . . . . . . 7 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
109xpeq1i 4392 . . . . . 6 (dom (𝐴𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V)
11 xpindir 4499 . . . . . 6 ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
1210, 11eqtri 2076 . . . . 5 (dom (𝐴𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V))
1312ineq2i 3162 . . . 4 (𝐴 ∩ (dom (𝐴𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
148, 13eqtri 2076 . . 3 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V)))
15 df-res 4384 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
167, 14, 153eqtr4i 2086 . 2 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
17 rescnvcnv 4810 . 2 (𝐴𝐵) = (𝐴𝐵)
1816, 17eqtri 2076 1 (𝐴 ↾ dom (𝐴𝐵)) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  Vcvv 2574  cin 2943   × cxp 4370  ccnv 4371  dom cdm 4372  cres 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384
This theorem is referenced by:  imadmres  4840
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