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Theorem resres 4652
Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
resres ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))

Proof of Theorem resres
StepHypRef Expression
1 df-res 4385 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
2 df-res 4385 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
32ineq1i 3162 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4 xpindir 4500 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
54ineq2i 3163 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6 df-res 4385 . . 3 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
7 inass 3175 . . 3 ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
85, 6, 73eqtr4ri 2087 . 2 ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵𝐶))
91, 3, 83eqtri 2080 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  Vcvv 2574  cin 2944   × cxp 4371  cres 4375
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 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-rel 4380  df-res 4385
This theorem is referenced by:  rescom  4664  resabs1  4668  resima2  4672  resmpt3  4685  resdisj  4779  rescnvcnv  4811  funimaexg  5011  fresin  5096  resdif  5176
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