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Theorem resindi 4716
Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
Assertion
Ref Expression
resindi (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem resindi
StepHypRef Expression
1 xpindir 4560 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
21ineq2i 3196 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
3 inindi 3215 . . 3 (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
42, 3eqtri 2108 . 2 (𝐴 ∩ ((𝐵𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
5 df-res 4440 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
6 df-res 4440 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 4440 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7ineq12i 3197 . 2 ((𝐴𝐵) ∩ (𝐴𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
94, 5, 83eqtr4i 2118 1 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1289  Vcvv 2619  cin 2996   × cxp 4426  cres 4430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434  df-rel 4435  df-res 4440
This theorem is referenced by:  resindm  4741
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