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Theorem afvpcfv0 27334
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 27320 . . 3  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
21eqeq1i 2365 . 2  |-  ( ( F''' A )  =  _V  <->  if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V )
3 eqcom 2360 . . . 4  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  _V  =  if ( F defAt  A ,  ( F `  A ) ,  _V ) )
4 eqif 3674 . . . 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 27312 . . . . . 6  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  (/) )
76eqcoms 2361 . . . . 5  |-  ( _V  =  ( F `  A )  ->  ( F `  A )  =  (/) )
87adantl 452 . . . 4  |-  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  -> 
( F `  A
)  =  (/) )
9 fvfundmfvn0 27311 . . . . . . 7  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 27297 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
119, 10sylibr 203 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  F defAt  A )
1211necon1bi 2564 . . . . 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 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864   (/)c0 3531   ifcif 3641   {csn 3716   dom cdm 4768    |` cres 4770   Fun wfun 5328   ` cfv 5334   defAt wdfat 27294  '''cafv 27295
This theorem is referenced by:  afvfv0bi  27340  aovpcov0  27378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-res 4780  df-iota 5298  df-fun 5336  df-fv 5342  df-dfat 27297  df-afv 27298
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