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Theorem fvresval 30416
Description: The value of a function at a restriction is either null or the same as the function itself. (Contributed by Scott Fenton, 4-Sep-2011.)
Assertion
Ref Expression
fvresval  |-  ( ( ( F  |`  B ) `
 A )  =  ( F `  A
)  \/  ( ( F  |`  B ) `  A )  =  (/) )

Proof of Theorem fvresval
StepHypRef Expression
1 exmid 416 . 2  |-  ( A  e.  B  \/  -.  A  e.  B )
2 fvres 5896 . . 3  |-  ( A  e.  B  ->  (
( F  |`  B ) `
 A )  =  ( F `  A
) )
3 nfvres 5912 . . 3  |-  ( -.  A  e.  B  -> 
( ( F  |`  B ) `  A
)  =  (/) )
42, 3orim12i 518 . 2  |-  ( ( A  e.  B  \/  -.  A  e.  B
)  ->  ( (
( F  |`  B ) `
 A )  =  ( F `  A
)  \/  ( ( F  |`  B ) `  A )  =  (/) ) )
51, 4ax-mp 5 1  |-  ( ( ( F  |`  B ) `
 A )  =  ( F `  A
)  \/  ( ( F  |`  B ) `  A )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    = wceq 1437    e. wcel 1872   (/)c0 3761    |` cres 4855   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-xp 4859  df-dm 4863  df-res 4865  df-iota 5565  df-fv 5609
This theorem is referenced by:  sltres  30559
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