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Theorem fvsnun2 5738
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5737. (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 4924 . . . 4  |-  ( G  |`  ( C  \  { A } ) )  =  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C 
\  { A }
) ) )  |`  ( C  \  { A } ) )
3 resundir 4942 . . . 4  |-  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  ( C  \  { A } ) )  =  ( ( { <. A ,  B >. }  |`  ( C  \  { A }
) )  u.  (
( F  |`  ( C  \  { A }
) )  |`  ( C  \  { A }
) ) )
4 disjdif 3510 . . . . . . 7  |-  ( { A }  i^i  ( C  \  { A }
) )  =  (/)
5 fvsnun.1 . . . . . . . . 9  |-  A  e. 
_V
6 fvsnun.2 . . . . . . . . 9  |-  B  e. 
_V
75, 6fnsn 5292 . . . . . . . 8  |-  { <. A ,  B >. }  Fn  { A }
8 fnresdisj 5348 . . . . . . . 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 145 . . . . . 6  |-  ( {
<. A ,  B >. }  |`  ( C  \  { A } ) )  =  (/)
11 residm 4960 . . . . . 6  |-  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) )  =  ( F  |`  ( C  \  { A } ) )
1210, 11uneq12i 3302 . . . . 5  |-  ( ( { <. A ,  B >. }  |`  ( C  \  { A } ) )  u.  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) ) )  =  (
(/)  u.  ( F  |`  ( C  \  { A } ) ) )
13 uncom 3294 . . . . 5  |-  ( (/)  u.  ( F  |`  ( C  \  { A }
) ) )  =  ( ( F  |`  ( C  \  { A } ) )  u.  (/) )
14 un0 3471 . . . . 5  |-  ( ( F  |`  ( C  \  { A } ) )  u.  (/) )  =  ( F  |`  ( C  \  { A }
) )
1512, 13, 143eqtri 2214 . . . 4  |-  ( ( { <. A ,  B >. }  |`  ( C  \  { A } ) )  u.  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) ) )  =  ( F  |`  ( C  \  { A } ) )
162, 3, 153eqtri 2214 . . 3  |-  ( G  |`  ( C  \  { A } ) )  =  ( F  |`  ( C  \  { A }
) )
1716fveq1i 5538 . 2  |-  ( ( G  |`  ( C  \  { A } ) ) `  D )  =  ( ( F  |`  ( C  \  { A } ) ) `  D )
18 fvres 5561 . 2  |-  ( D  e.  ( C  \  { A } )  -> 
( ( G  |`  ( C  \  { A } ) ) `  D )  =  ( G `  D ) )
19 fvres 5561 . 2  |-  ( D  e.  ( C  \  { A } )  -> 
( ( F  |`  ( C  \  { A } ) ) `  D )  =  ( F `  D ) )
2017, 18, 193eqtr3a 2246 1  |-  ( D  e.  ( C  \  { A } )  -> 
( G `  D
)  =  ( F `
 D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 105    = wceq 1364    e. wcel 2160   _Vcvv 2752    \ cdif 3141    u. cun 3142    i^i cin 3143   (/)c0 3437   {csn 3610   <.cop 3613    |` cres 4649    Fn wfn 5233   ` cfv 5238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-res 4659  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246
This theorem is referenced by:  facnn  10748
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