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Theorem afvpcfv0 38051
Description: If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvpcfv0  |-  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) )

Proof of Theorem afvpcfv0
StepHypRef Expression
1 dfafv2 38037 . . 3  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
21eqeq1i 2427 . 2  |-  ( ( F''' A )  =  _V  <->  if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V )
3 eqcom 2429 . . . 4  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  _V  =  if ( F defAt  A ,  ( F `  A ) ,  _V ) )
4 eqif 3944 . . . 4  |-  ( _V  =  if ( F defAt 
A ,  ( F `
 A ) ,  _V )  <->  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V ) ) )
53, 4bitri 252 . . 3  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  ( ( F defAt 
A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V ) ) )
6 fveqvfvv 38029 . . . . . 6  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  (/) )
76eqcoms 2432 . . . . 5  |-  ( _V  =  ( F `  A )  ->  ( F `  A )  =  (/) )
87adantl 467 . . . 4  |-  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  -> 
( F `  A
)  =  (/) )
9 fvfundmfvn0 5904 . . . . . . 7  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 38021 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
119, 10sylibr 215 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  F defAt  A )
1211necon1bi 2655 . . . . 5  |-  ( -.  F defAt  A  ->  ( F `  A )  =  (/) )
1312adantr 466 . . . 4  |-  ( ( -.  F defAt  A  /\  _V  =  _V )  ->  ( F `  A
)  =  (/) )
148, 13jaoi 380 . . 3  |-  ( ( ( F defAt  A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V )
)  ->  ( F `  A )  =  (/) )
155, 14sylbi 198 . 2  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  ->  ( F `  A )  =  (/) )
162, 15sylbi 198 1  |-  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1867    =/= wne 2616   _Vcvv 3078   (/)c0 3758   ifcif 3906   {csn 3993   dom cdm 4845    |` cres 4847   Fun wfun 5586   ` cfv 5592   defAt wdfat 38018  '''cafv 38019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5556  df-fun 5594  df-fv 5600  df-dfat 38021  df-afv 38022
This theorem is referenced by:  afvfv0bi  38057  aovpcov0  38095
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