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Theorem fvunsn 6104
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 5298 . . . 4  |-  ( ( A  u.  { <. B ,  C >. } )  |`  { D } )  =  ( ( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )
2 elsni 4057 . . . . . . . 8  |-  ( B  e.  { D }  ->  B  =  D )
32necon3ai 2685 . . . . . . 7  |-  ( B  =/=  D  ->  -.  B  e.  { D } )
4 ressnop0 6079 . . . . . . 7  |-  ( -.  B  e.  { D }  ->  ( { <. B ,  C >. }  |`  { D } )  =  (/) )
53, 4syl 16 . . . . . 6  |-  ( B  =/=  D  ->  ( { <. B ,  C >. }  |`  { D } )  =  (/) )
65uneq2d 3654 . . . . 5  |-  ( B  =/=  D  ->  (
( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )  =  ( ( A  |`  { D } )  u.  (/) ) )
7 un0 3819 . . . . 5  |-  ( ( A  |`  { D } )  u.  (/) )  =  ( A  |`  { D } )
86, 7syl6eq 2514 . . . 4  |-  ( B  =/=  D  ->  (
( A  |`  { D } )  u.  ( { <. B ,  C >. }  |`  { D } ) )  =  ( A  |`  { D } ) )
91, 8syl5eq 2510 . . 3  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } )  |`  { D } )  =  ( A  |`  { D } ) )
109fveq1d 5874 . 2  |-  ( B  =/=  D  ->  (
( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  |`  { D } ) `  D
) )
11 fvressn 6088 . . 3  |-  ( D  e.  _V  ->  (
( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D ) )
12 fvprc 5866 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  (/) )
13 fvprc 5866 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( A  u.  { <. B ,  C >. } ) `  D )  =  (/) )
1412, 13eqtr4d 2501 . . 3  |-  ( -.  D  e.  _V  ->  ( ( ( A  u.  {
<. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D ) )
1511, 14pm2.61i 164 . 2  |-  ( ( ( A  u.  { <. B ,  C >. } )  |`  { D } ) `  D
)  =  ( ( A  u.  { <. B ,  C >. } ) `
 D )
16 fvressn 6088 . . 3  |-  ( D  e.  _V  ->  (
( A  |`  { D } ) `  D
)  =  ( A `
 D ) )
17 fvprc 5866 . . . 4  |-  ( -.  D  e.  _V  ->  ( ( A  |`  { D } ) `  D
)  =  (/) )
18 fvprc 5866 . . . 4  |-  ( -.  D  e.  _V  ->  ( A `  D )  =  (/) )
1917, 18eqtr4d 2501 . . 3  |-  ( -.  D  e.  _V  ->  ( ( A  |`  { D } ) `  D
)  =  ( A `
 D ) )
2016, 19pm2.61i 164 . 2  |-  ( ( A  |`  { D } ) `  D
)  =  ( A `
 D )
2110, 15, 203eqtr3g 2521 1  |-  ( B  =/=  D  ->  (
( A  u.  { <. B ,  C >. } ) `  D )  =  ( A `  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109    u. cun 3469   (/)c0 3793   {csn 4032   <.cop 4038    |` cres 5010   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-res 5020  df-iota 5557  df-fv 5602
This theorem is referenced by:  fvpr1  6115  fvpr1g  6117  fvpr2g  6118  fvtp1  6119  fvtp1g  6122  ac6sfi  7782  cats1un  12712  ruclem6  13979  ruclem7  13980  eupap1  25102  fnchoice  31565
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