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Theorem fvunsn 5712
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 4970 . . . 4  |-  ( ( A  u.  { <. B ,  C >. } )  |`  { D } )  =  ( ( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )
2 elsni 3664 . . . . . . . 8  |-  ( B  e.  { D }  ->  B  =  D )
32necon3ai 2486 . . . . . . 7  |-  ( B  =/=  D  ->  -.  B  e.  { D } )
4 ressnop0 5703 . . . . . . 7  |-  ( -.  B  e.  { D }  ->  ( { <. B ,  C >. }  |`  { D } )  =  (/) )
53, 4syl 15 . . . . . 6  |-  ( B  =/=  D  ->  ( { <. B ,  C >. }  |`  { D } )  =  (/) )
65uneq2d 3329 . . . . 5  |-  ( B  =/=  D  ->  (
( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )  =  ( ( A  |`  { D } )  u.  (/) ) )
7 un0 3479 . . . . 5  |-  ( ( A  |`  { D } )  u.  (/) )  =  ( A  |`  { D } )
86, 7syl6eq 2331 . . . 4  |-  ( B  =/=  D  ->  (
( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )  =  ( A  |`  { D } ) )
91, 8syl5eq 2327 . . 3  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } )  |`  { D } )  =  ( A  |`  { D } ) )
109fveq1d 5527 . 2  |-  ( B  =/=  D  ->  (
( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  |`  { D } ) `  D
) )
11 snidg 3665 . . . 4  |-  ( D  e.  _V  ->  D  e.  { D } )
12 fvres 5542 . . . 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 5519 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  (/) )
15 fvprc 5519 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( A  u.  { <. B ,  C >. } ) `  D )  =  (/) )
1614, 15eqtr4d 2318 . . 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 5542 . . . 4  |-  ( D  e.  { D }  ->  ( ( A  |`  { D } ) `  D )  =  ( A `  D ) )
1911, 18syl 15 . . 3  |-  ( D  e.  _V  ->  (
( A  |`  { D } ) `  D
)  =  ( A `
 D ) )
20 fvprc 5519 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( A  |`  { D } ) `  D
)  =  (/) )
21 fvprc 5519 . . . 4  |-  ( -.  D  e.  _V  ->  ( A `  D )  =  (/) )
2220, 21eqtr4d 2318 . . 3  |-  ( -.  D  e.  _V  ->  ( ( A  |`  { D } ) `  D
)  =  ( A `
 D ) )
2319, 22pm2.61i 156 . 2  |-  ( ( A  |`  { D } ) `  D
)  =  ( A `
 D )
2410, 17, 233eqtr3g 2338 1  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } ) `  D )  =  ( A `  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    u. cun 3150   (/)c0 3455   {csn 3640   <.cop 3643    |` cres 4691   ` cfv 5255
This theorem is referenced by:  fvpr1  5722  fvtp1  5724  ac6sfi  7101  cats1un  11476  ruclem6  12513  ruclem7  12514  eupap1  23900  fnchoice  27700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-res 4701  df-iota 5219  df-fv 5263
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