Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvpcfv0 Unicode version

Theorem afvpcfv0 28009
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 27995 . . 3  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
21eqeq1i 2290 . 2  |-  ( ( F''' A )  =  _V  <->  if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V )
3 eqcom 2285 . . . 4  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  _V  =  if ( F defAt  A ,  ( F `  A ) ,  _V ) )
4 eqif 3598 . . . 4  |-  ( _V  =  if ( F defAt 
A ,  ( F `
 A ) ,  _V )  <->  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V ) ) )
53, 4bitri 240 . . 3  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  ( ( F defAt 
A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V ) ) )
6 fveqvfvv 27987 . . . . . 6  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  (/) )
76eqcoms 2286 . . . . 5  |-  ( _V  =  ( F `  A )  ->  ( F `  A )  =  (/) )
87adantl 452 . . . 4  |-  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  -> 
( F `  A
)  =  (/) )
9 fvfundmfvn0 27986 . . . . . . 7  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 27974 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
119, 10sylibr 203 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  F defAt  A )
1211necon1bi 2489 . . . . 5  |-  ( -.  F defAt  A  ->  ( F `  A )  =  (/) )
1312adantr 451 . . . 4  |-  ( ( -.  F defAt  A  /\  _V  =  _V )  ->  ( F `  A
)  =  (/) )
148, 13jaoi 368 . . 3  |-  ( ( ( F defAt  A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V )
)  ->  ( F `  A )  =  (/) )
155, 14sylbi 187 . 2  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  ->  ( F `  A )  =  (/) )
162, 15sylbi 187 1  |-  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   ifcif 3565   {csn 3640   dom cdm 4689    |` cres 4691   Fun wfun 5249   ` cfv 5255   defAt wdfat 27971  '''cafv 27972
This theorem is referenced by:  afvfv0bi  28015  aovpcov0  28050
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-sbc 2992  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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-dfat 27974  df-afv 27975
  Copyright terms: Public domain W3C validator