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Theorem fvsnun2 6116
Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 6115. (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 5107 . . . 4  |-  ( G  |`  ( C  \  { A } ) )  =  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C 
\  { A }
) ) )  |`  ( C  \  { A } ) )
3 resundir 5125 . . . 4  |-  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  ( C  \  { A } ) )  =  ( ( { <. A ,  B >. }  |`  ( C  \  { A }
) )  u.  (
( F  |`  ( C  \  { A }
) )  |`  ( C  \  { A }
) ) )
4 disjdif 3830 . . . . . . 7  |-  ( { A }  i^i  ( C  \  { A }
) )  =  (/)
5 fvsnun.1 . . . . . . . . 9  |-  A  e. 
_V
6 fvsnun.2 . . . . . . . . 9  |-  B  e. 
_V
75, 6fnsn 5642 . . . . . . . 8  |-  { <. A ,  B >. }  Fn  { A }
8 fnresdisj 5696 . . . . . . . 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 213 . . . . . 6  |-  ( {
<. A ,  B >. }  |`  ( C  \  { A } ) )  =  (/)
11 residm 5142 . . . . . 6  |-  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) )  =  ( F  |`  ( C  \  { A } ) )
1210, 11uneq12i 3577 . . . . 5  |-  ( ( { <. A ,  B >. }  |`  ( C  \  { A } ) )  u.  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) ) )  =  (
(/)  u.  ( F  |`  ( C  \  { A } ) ) )
13 uncom 3569 . . . . 5  |-  ( (/)  u.  ( F  |`  ( C  \  { A }
) ) )  =  ( ( F  |`  ( C  \  { A } ) )  u.  (/) )
14 un0 3762 . . . . 5  |-  ( ( F  |`  ( C  \  { A } ) )  u.  (/) )  =  ( F  |`  ( C  \  { A }
) )
1512, 13, 143eqtri 2497 . . . 4  |-  ( ( { <. A ,  B >. }  |`  ( C  \  { A } ) )  u.  ( ( F  |`  ( C  \  { A } ) )  |`  ( C  \  { A } ) ) )  =  ( F  |`  ( C  \  { A } ) )
162, 3, 153eqtri 2497 . . 3  |-  ( G  |`  ( C  \  { A } ) )  =  ( F  |`  ( C  \  { A }
) )
1716fveq1i 5880 . 2  |-  ( ( G  |`  ( C  \  { A } ) ) `  D )  =  ( ( F  |`  ( C  \  { A } ) ) `  D )
18 fvres 5893 . 2  |-  ( D  e.  ( C  \  { A } )  -> 
( ( G  |`  ( C  \  { A } ) ) `  D )  =  ( G `  D ) )
19 fvres 5893 . 2  |-  ( D  e.  ( C  \  { A } )  -> 
( ( F  |`  ( C  \  { A } ) ) `  D )  =  ( F `  D ) )
2017, 18, 193eqtr3a 2529 1  |-  ( D  e.  ( C  \  { A } )  -> 
( G `  D
)  =  ( F `
 D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    = wceq 1452    e. wcel 1904   _Vcvv 3031    \ cdif 3387    u. cun 3388    i^i cin 3389   (/)c0 3722   {csn 3959   <.cop 3965    |` cres 4841    Fn wfn 5584   ` cfv 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597
This theorem is referenced by:  facnn  12499
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