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Theorem resdisj 5290
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 5151 . 2  |-  ( ( C  |`  A )  |`  B )  =  ( C  |`  ( A  i^i  B ) )
2 reseq2 5133 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  |`  ( A  i^i  B
) )  =  ( C  |`  (/) ) )
3 res0 5142 . . 3  |-  ( C  |`  (/) )  =  (/)
42, 3syl6eq 2483 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  |`  ( A  i^i  B
) )  =  (/) )
51, 4syl5eq 2479 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  |`  A )  |`  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    i^i cin 3311   (/)c0 3620    |` cres 4872
This theorem is referenced by:  fvsnun1  5920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-opab 4259  df-xp 4876  df-rel 4877  df-res 4882
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