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Theorem resdisj 5373
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 5225 . 2  |-  ( ( C  |`  A )  |`  B )  =  ( C  |`  ( A  i^i  B ) )
2 reseq2 5208 . . 3  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  |`  ( A  i^i  B
) )  =  ( C  |`  (/) ) )
3 res0 5217 . . 3  |-  ( C  |`  (/) )  =  (/)
42, 3syl6eq 2457 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( C  |`  ( A  i^i  B
) )  =  (/) )
51, 4syl5eq 2453 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( ( C  |`  A )  |`  B )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1403    i^i cin 3410   (/)c0 3735    |` cres 4942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pr 4627
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-opab 4451  df-xp 4946  df-rel 4947  df-res 4952
This theorem is referenced by:  fvsnun1  6040
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