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Theorem resdisj 5105
Description: A double restriction to disjoint classes is the empty set. (Contributed by NM, 7-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
resdisj  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  |`  A )  |`  B )  =  (/) )

Proof of Theorem resdisj
StepHypRef Expression
1 resres 4968 . 2  |-  ( ( C  |`  A )  |`  B )  =  ( C  |`  ( A  i^i  B ) )
2 reseq2 4950 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  |`  ( A  i^i  B
) )  =  ( C  |`  (/) ) )
3 res0 4959 . . 3  |-  ( C  |`  (/) )  =  (/)
42, 3syl6eq 2331 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  |`  ( A  i^i  B
) )  =  (/) )
51, 4syl5eq 2327 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  |`  A )  |`  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    i^i cin 3151   (/)c0 3455    |` cres 4691
This theorem is referenced by:  fvsnun1  5715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-opab 4078  df-xp 4695  df-rel 4696  df-res 4701
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