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Theorem fvsnun2 5611
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5610. (Contributed by NM, 23-Sep-2007.)
Hypotheses
Ref Expression
fvsnun.1  |-  A  e. 
_V
fvsnun.2  |-  B  e. 
_V
fvsnun.3  |-  G  =  ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )
Assertion
Ref Expression
fvsnun2  |-  ( D  e.  ( C  \  { A } )  -> 
( G `  D
)  =  ( F `
 D ) )

Proof of Theorem fvsnun2
StepHypRef Expression
1 fvsnun.3 . . . . 5  |-  G  =  ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )
21reseq1i 4810 . . . 4  |-  ( G  |`  ( C  \  { A } ) )  =  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C 
\  { A }
) ) )  |`  ( C  \  { A } ) )
3 resundir 4828 . . . 4  |-  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  ( C  \  { A } ) )  =  ( ( { <. A ,  B >. }  |`  ( C  \  { A }
) )  u.  (
( F  |`  ( C  \  { A }
) )  |`  ( C  \  { A }
) ) )
4 disjdif 3430 . . . . . . 7  |-  ( { A }  i^i  ( C  \  { A }
) )  =  (/)
5 fvsnun.1 . . . . . . . . 9  |-  A  e. 
_V
6 fvsnun.2 . . . . . . . . 9  |-  B  e. 
_V
75, 6fnsn 5172 . . . . . . . 8  |-  { <. A ,  B >. }  Fn  { A }
8 fnresdisj 5228 . . . . . . . 8  |-  ( {
<. A ,  B >. }  Fn  { A }  ->  ( ( { A }  i^i  ( C  \  { A } ) )  =  (/)  <->  ( { <. A ,  B >. }  |`  ( C  \  { A }
) )  =  (/) ) )
97, 8ax-mp 5 . . . . . . 7  |-  ( ( { A }  i^i  ( C  \  { A } ) )  =  (/) 
<->  ( { <. A ,  B >. }  |`  ( C  \  { A }
) )  =  (/) )
104, 9mpbi 144 . . . . . 6  |-  ( {
<. A ,  B >. }  |`  ( C  \  { A } ) )  =  (/)
11 residm 4846 . . . . . 6  |-  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) )  =  ( F  |`  ( C  \  { A } ) )
1210, 11uneq12i 3223 . . . . 5  |-  ( ( { <. A ,  B >. }  |`  ( C  \  { A } ) )  u.  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) ) )  =  (
(/)  u.  ( F  |`  ( C  \  { A } ) ) )
13 uncom 3215 . . . . 5  |-  ( (/)  u.  ( F  |`  ( C  \  { A }
) ) )  =  ( ( F  |`  ( C  \  { A } ) )  u.  (/) )
14 un0 3391 . . . . 5  |-  ( ( F  |`  ( C  \  { A } ) )  u.  (/) )  =  ( F  |`  ( C  \  { A }
) )
1512, 13, 143eqtri 2162 . . . 4  |-  ( ( { <. A ,  B >. }  |`  ( C  \  { A } ) )  u.  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) ) )  =  ( F  |`  ( C  \  { A } ) )
162, 3, 153eqtri 2162 . . 3  |-  ( G  |`  ( C  \  { A } ) )  =  ( F  |`  ( C  \  { A }
) )
1716fveq1i 5415 . 2  |-  ( ( G  |`  ( C  \  { A } ) ) `  D )  =  ( ( F  |`  ( C  \  { A } ) ) `  D )
18 fvres 5438 . 2  |-  ( D  e.  ( C  \  { A } )  -> 
( ( G  |`  ( C  \  { A } ) ) `  D )  =  ( G `  D ) )
19 fvres 5438 . 2  |-  ( D  e.  ( C  \  { A } )  -> 
( ( F  |`  ( C  \  { A } ) ) `  D )  =  ( F `  D ) )
2017, 18, 193eqtr3a 2194 1  |-  ( D  e.  ( C  \  { A } )  -> 
( G `  D
)  =  ( F `
 D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   _Vcvv 2681    \ cdif 3063    u. cun 3064    i^i cin 3065   (/)c0 3358   {csn 3522   <.cop 3525    |` cres 4536    Fn wfn 5113   ` cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126
This theorem is referenced by:  facnn  10466
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