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Theorem resdisj 5378
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 5234 . 2  |-  ( ( C  |`  A )  |`  B )  =  ( C  |`  ( A  i^i  B ) )
2 reseq2 5216 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  |`  ( A  i^i  B
) )  =  ( C  |`  (/) ) )
3 res0 5226 . . 3  |-  ( C  |`  (/) )  =  (/)
42, 3syl6eq 2511 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  |`  ( A  i^i  B
) )  =  (/) )
51, 4syl5eq 2507 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  |`  A )  |`  B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    i^i cin 3438   (/)c0 3748    |` cres 4953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-opab 4462  df-xp 4957  df-rel 4958  df-res 4963
This theorem is referenced by:  fvsnun1  6025
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