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Theorem fvsnun1 6025
Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 6026. (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 5217 . . . 4  |-  ( G  |`  { A } )  =  ( ( {
<. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  { A } )
3 resundir 5236 . . . . 5  |-  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  { A } )  =  ( ( {
<. A ,  B >. }  |`  { A } )  u.  ( ( F  |`  ( C  \  { A } ) )  |`  { A } ) )
4 incom 3654 . . . . . . . . 9  |-  ( ( C  \  { A } )  i^i  { A } )  =  ( { A }  i^i  ( C  \  { A } ) )
5 disjdif 3862 . . . . . . . . 9  |-  ( { A }  i^i  ( C  \  { A }
) )  =  (/)
64, 5eqtri 2483 . . . . . . . 8  |-  ( ( C  \  { A } )  i^i  { A } )  =  (/)
7 resdisj 5378 . . . . . . . 8  |-  ( ( ( C  \  { A } )  i^i  { A } )  =  (/)  ->  ( ( F  |`  ( C  \  { A } ) )  |`  { A } )  =  (/) )
86, 7ax-mp 5 . . . . . . 7  |-  ( ( F  |`  ( C  \  { A } ) )  |`  { A } )  =  (/)
98uneq2i 3618 . . . . . 6  |-  ( ( { <. A ,  B >. }  |`  { A } )  u.  (
( F  |`  ( C  \  { A }
) )  |`  { A } ) )  =  ( ( { <. A ,  B >. }  |`  { A } )  u.  (/) )
10 un0 3773 . . . . . 6  |-  ( ( { <. A ,  B >. }  |`  { A } )  u.  (/) )  =  ( { <. A ,  B >. }  |`  { A } )
119, 10eqtri 2483 . . . . 5  |-  ( ( { <. A ,  B >. }  |`  { A } )  u.  (
( F  |`  ( C  \  { A }
) )  |`  { A } ) )  =  ( { <. A ,  B >. }  |`  { A } )
123, 11eqtri 2483 . . . 4  |-  ( ( { <. A ,  B >. }  u.  ( F  |`  ( C  \  { A } ) ) )  |`  { A } )  =  ( { <. A ,  B >. }  |`  { A } )
132, 12eqtri 2483 . . 3  |-  ( G  |`  { A } )  =  ( { <. A ,  B >. }  |`  { A } )
1413fveq1i 5803 . 2  |-  ( ( G  |`  { A } ) `  A
)  =  ( ( { <. A ,  B >. }  |`  { A } ) `  A
)
15 fvsnun.1 . . . 4  |-  A  e. 
_V
1615snid 4016 . . 3  |-  A  e. 
{ A }
17 fvres 5816 . . 3  |-  ( A  e.  { A }  ->  ( ( G  |`  { A } ) `  A )  =  ( G `  A ) )
1816, 17ax-mp 5 . 2  |-  ( ( G  |`  { A } ) `  A
)  =  ( G `
 A )
19 fvres 5816 . . . 4  |-  ( A  e.  { A }  ->  ( ( { <. A ,  B >. }  |`  { A } ) `  A
)  =  ( {
<. A ,  B >. } `
 A ) )
2016, 19ax-mp 5 . . 3  |-  ( ( { <. A ,  B >. }  |`  { A } ) `  A
)  =  ( {
<. A ,  B >. } `
 A )
21 fvsnun.2 . . . 4  |-  B  e. 
_V
2215, 21fvsn 6023 . . 3  |-  ( {
<. A ,  B >. } `
 A )  =  B
2320, 22eqtri 2483 . 2  |-  ( ( { <. A ,  B >. }  |`  { A } ) `  A
)  =  B
2414, 18, 233eqtr3i 2491 1  |-  ( G `
 A )  =  B
Colors of variables: wff setvar class
Syntax hints:    = wceq 1370    e. wcel 1758   _Vcvv 3078    \ cdif 3436    u. cun 3437    i^i cin 3438   (/)c0 3748   {csn 3988   <.cop 3994    |` cres 4953   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-res 4963  df-iota 5492  df-fun 5531  df-fv 5537
This theorem is referenced by:  fac0  12175  ruclem4  13638
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