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Theorem resindi 4983
 Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
Assertion
Ref Expression
resindi (A (BC)) = ((A B) ∩ (A C))

Proof of Theorem resindi
StepHypRef Expression
1 xpindir 4865 . . . 4 ((BC) × V) = ((B × V) ∩ (C × V))
21ineq2i 3454 . . 3 (A ∩ ((BC) × V)) = (A ∩ ((B × V) ∩ (C × V)))
3 inindi 3472 . . 3 (A ∩ ((B × V) ∩ (C × V))) = ((A ∩ (B × V)) ∩ (A ∩ (C × V)))
42, 3eqtri 2373 . 2 (A ∩ ((BC) × V)) = ((A ∩ (B × V)) ∩ (A ∩ (C × V)))
5 df-res 4788 . 2 (A (BC)) = (A ∩ ((BC) × V))
6 df-res 4788 . . 3 (A B) = (A ∩ (B × V))
7 df-res 4788 . . 3 (A C) = (A ∩ (C × V))
86, 7ineq12i 3455 . 2 ((A B) ∩ (A C)) = ((A ∩ (B × V)) ∩ (A ∩ (C × V)))
94, 5, 83eqtr4i 2383 1 (A (BC)) = ((A B) ∩ (A C))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∩ cin 3208   × cxp 4770   ↾ cres 4774 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-xp 4784  df-res 4788 This theorem is referenced by: (None)
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