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Theorem fvunsn 5925
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 5161 . . . 4  |-  ( ( A  u.  { <. B ,  C >. } )  |`  { D } )  =  ( ( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )
2 elsni 3838 . . . . . . . 8  |-  ( B  e.  { D }  ->  B  =  D )
32necon3ai 2644 . . . . . . 7  |-  ( B  =/=  D  ->  -.  B  e.  { D } )
4 ressnop0 5913 . . . . . . 7  |-  ( -.  B  e.  { D }  ->  ( { <. B ,  C >. }  |`  { D } )  =  (/) )
53, 4syl 16 . . . . . 6  |-  ( B  =/=  D  ->  ( { <. B ,  C >. }  |`  { D } )  =  (/) )
65uneq2d 3501 . . . . 5  |-  ( B  =/=  D  ->  (
( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )  =  ( ( A  |`  { D } )  u.  (/) ) )
7 un0 3652 . . . . 5  |-  ( ( A  |`  { D } )  u.  (/) )  =  ( A  |`  { D } )
86, 7syl6eq 2484 . . . 4  |-  ( B  =/=  D  ->  (
( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )  =  ( A  |`  { D } ) )
91, 8syl5eq 2480 . . 3  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } )  |`  { D } )  =  ( A  |`  { D } ) )
109fveq1d 5730 . 2  |-  ( B  =/=  D  ->  (
( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  |`  { D } ) `  D
) )
11 snidg 3839 . . . 4  |-  ( D  e.  _V  ->  D  e.  { D } )
12 fvres 5745 . . . 4  |-  ( D  e.  { D }  ->  ( ( ( A  u.  { <. B ,  C >. } )  |`  { D } ) `  D )  =  ( ( A  u.  { <. B ,  C >. } ) `  D ) )
1311, 12syl 16 . . 3  |-  ( D  e.  _V  ->  (
( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D ) )
14 fvprc 5722 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  (/) )
15 fvprc 5722 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( A  u.  { <. B ,  C >. } ) `  D )  =  (/) )
1614, 15eqtr4d 2471 . . 3  |-  ( -.  D  e.  _V  ->  ( ( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D ) )
1713, 16pm2.61i 158 . 2  |-  ( ( ( A  u.  { <. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D )
18 fvres 5745 . . . 4  |-  ( D  e.  { D }  ->  ( ( A  |`  { D } ) `  D )  =  ( A `  D ) )
1911, 18syl 16 . . 3  |-  ( D  e.  _V  ->  (
( A  |`  { D } ) `  D
)  =  ( A `
 D ) )
20 fvprc 5722 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( A  |`  { D } ) `  D
)  =  (/) )
21 fvprc 5722 . . . 4  |-  ( -.  D  e.  _V  ->  ( A `  D )  =  (/) )
2220, 21eqtr4d 2471 . . 3  |-  ( -.  D  e.  _V  ->  ( ( A  |`  { D } ) `  D
)  =  ( A `
 D ) )
2319, 22pm2.61i 158 . 2  |-  ( ( A  |`  { D } ) `  D
)  =  ( A `
 D )
2410, 17, 233eqtr3g 2491 1  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } ) `  D )  =  ( A `  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    u. cun 3318   (/)c0 3628   {csn 3814   <.cop 3817    |` cres 4880   ` cfv 5454
This theorem is referenced by:  fvpr1  5935  fvpr1g  5937  fvpr2g  5938  fvtp1  5939  fvtp1g  5942  ac6sfi  7351  cats1un  11790  ruclem6  12834  ruclem7  12835  eupap1  21698  fnchoice  27676
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-res 4890  df-iota 5418  df-fv 5462
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