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Theorem fvsnun1 5731
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 5732. (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
fvsnun1  |-  ( G `
 A )  =  B

Proof of Theorem fvsnun1
StepHypRef Expression
1 fvsnun.3 . . . . 5  |-  G  =  ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )
21reseq1i 4967 . . . 4  |-  ( G  |`  { A } )  =  ( ( {
<. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  { A } )
3 resundir 4986 . . . . 5  |-  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  { A } )  =  ( ( {
<. A ,  B >. }  |`  { A } )  u.  ( ( F  |`  ( C  \  { A } ) )  |`  { A } ) )
4 incom 3374 . . . . . . . . 9  |-  ( ( C  \  { A } )  i^i  { A } )  =  ( { A }  i^i  ( C  \  { A } ) )
5 disjdif 3539 . . . . . . . . 9  |-  ( { A }  i^i  ( C  \  { A }
) )  =  (/)
64, 5eqtri 2316 . . . . . . . 8  |-  ( ( C  \  { A } )  i^i  { A } )  =  (/)
7 resdisj 5121 . . . . . . . 8  |-  ( ( ( C  \  { A } )  i^i  { A } )  =  (/)  ->  ( ( F  |`  ( C  \  { A } ) )  |`  { A } )  =  (/) )
86, 7ax-mp 8 . . . . . . 7  |-  ( ( F  |`  ( C  \  { A } ) )  |`  { A } )  =  (/)
98uneq2i 3339 . . . . . 6  |-  ( ( { <. A ,  B >. }  |`  { A } )  u.  (
( F  |`  ( C  \  { A }
) )  |`  { A } ) )  =  ( ( { <. A ,  B >. }  |`  { A } )  u.  (/) )
10 un0 3492 . . . . . 6  |-  ( ( { <. A ,  B >. }  |`  { A } )  u.  (/) )  =  ( { <. A ,  B >. }  |`  { A } )
119, 10eqtri 2316 . . . . 5  |-  ( ( { <. A ,  B >. }  |`  { A } )  u.  (
( F  |`  ( C  \  { A }
) )  |`  { A } ) )  =  ( { <. A ,  B >. }  |`  { A } )
123, 11eqtri 2316 . . . 4  |-  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  { A } )  =  ( { <. A ,  B >. }  |`  { A } )
132, 12eqtri 2316 . . 3  |-  ( G  |`  { A } )  =  ( { <. A ,  B >. }  |`  { A } )
1413fveq1i 5542 . 2  |-  ( ( G  |`  { A } ) `  A
)  =  ( ( { <. A ,  B >. }  |`  { A } ) `  A
)
15 fvsnun.1 . . . 4  |-  A  e. 
_V
1615snid 3680 . . 3  |-  A  e. 
{ A }
17 fvres 5558 . . 3  |-  ( A  e.  { A }  ->  ( ( G  |`  { A } ) `  A )  =  ( G `  A ) )
1816, 17ax-mp 8 . 2  |-  ( ( G  |`  { A } ) `  A
)  =  ( G `
 A )
19 fvres 5558 . . . 4  |-  ( A  e.  { A }  ->  ( ( { <. A ,  B >. }  |`  { A } ) `  A
)  =  ( {
<. A ,  B >. } `
 A ) )
2016, 19ax-mp 8 . . 3  |-  ( ( { <. A ,  B >. }  |`  { A } ) `  A
)  =  ( {
<. A ,  B >. } `
 A )
21 fvsnun.2 . . . 4  |-  B  e. 
_V
2215, 21fvsn 5729 . . 3  |-  ( {
<. A ,  B >. } `
 A )  =  B
2320, 22eqtri 2316 . 2  |-  ( ( { <. A ,  B >. }  |`  { A } ) `  A
)  =  B
2414, 18, 233eqtr3i 2324 1  |-  ( G `
 A )  =  B
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   <.cop 3656    |` cres 4707   ` cfv 5271
This theorem is referenced by:  fac0  11307  ruclem4  12528
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279
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