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Theorem fvunsn 5728
Description: Remove an ordered pair not participating in a function value. (Contributed by NM, 1-Oct-2013.) (Revised by Mario Carneiro, 28-May-2014.)
Assertion
Ref Expression
fvunsn  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } ) `  D )  =  ( A `  D ) )

Proof of Theorem fvunsn
StepHypRef Expression
1 resundir 4986 . . . 4  |-  ( ( A  u.  { <. B ,  C >. } )  |`  { D } )  =  ( ( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )
2 elsni 3677 . . . . . . . 8  |-  ( B  e.  { D }  ->  B  =  D )
32necon3ai 2499 . . . . . . 7  |-  ( B  =/=  D  ->  -.  B  e.  { D } )
4 ressnop0 5719 . . . . . . 7  |-  ( -.  B  e.  { D }  ->  ( { <. B ,  C >. }  |`  { D } )  =  (/) )
53, 4syl 15 . . . . . 6  |-  ( B  =/=  D  ->  ( { <. B ,  C >. }  |`  { D } )  =  (/) )
65uneq2d 3342 . . . . 5  |-  ( B  =/=  D  ->  (
( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )  =  ( ( A  |`  { D } )  u.  (/) ) )
7 un0 3492 . . . . 5  |-  ( ( A  |`  { D } )  u.  (/) )  =  ( A  |`  { D } )
86, 7syl6eq 2344 . . . 4  |-  ( B  =/=  D  ->  (
( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )  =  ( A  |`  { D } ) )
91, 8syl5eq 2340 . . 3  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } )  |`  { D } )  =  ( A  |`  { D } ) )
109fveq1d 5543 . 2  |-  ( B  =/=  D  ->  (
( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  |`  { D } ) `  D
) )
11 snidg 3678 . . . 4  |-  ( D  e.  _V  ->  D  e.  { D } )
12 fvres 5558 . . . 4  |-  ( D  e.  { D }  ->  ( ( ( A  u.  { <. B ,  C >. } )  |`  { D } ) `  D )  =  ( ( A  u.  { <. B ,  C >. } ) `  D ) )
1311, 12syl 15 . . 3  |-  ( D  e.  _V  ->  (
( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D ) )
14 fvprc 5535 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  (/) )
15 fvprc 5535 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( A  u.  { <. B ,  C >. } ) `  D )  =  (/) )
1614, 15eqtr4d 2331 . . 3  |-  ( -.  D  e.  _V  ->  ( ( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D ) )
1713, 16pm2.61i 156 . 2  |-  ( ( ( A  u.  { <. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D )
18 fvres 5558 . . . 4  |-  ( D  e.  { D }  ->  ( ( A  |`  { D } ) `  D )  =  ( A `  D ) )
1911, 18syl 15 . . 3  |-  ( D  e.  _V  ->  (
( A  |`  { D } ) `  D
)  =  ( A `
 D ) )
20 fvprc 5535 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( A  |`  { D } ) `  D
)  =  (/) )
21 fvprc 5535 . . . 4  |-  ( -.  D  e.  _V  ->  ( A `  D )  =  (/) )
2220, 21eqtr4d 2331 . . 3  |-  ( -.  D  e.  _V  ->  ( ( A  |`  { D } ) `  D
)  =  ( A `
 D ) )
2319, 22pm2.61i 156 . 2  |-  ( ( A  |`  { D } ) `  D
)  =  ( A `
 D )
2410, 17, 233eqtr3g 2351 1  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } ) `  D )  =  ( A `  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    u. cun 3163   (/)c0 3468   {csn 3653   <.cop 3656    |` cres 4707   ` cfv 5271
This theorem is referenced by:  fvpr1  5738  fvtp1  5740  ac6sfi  7117  cats1un  11492  ruclem6  12529  ruclem7  12530  eupap1  23915  fnchoice  27803  constr3lem4  28393
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-13 1698  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-pow 4204  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-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-xp 4711  df-res 4717  df-iota 5235  df-fv 5279
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