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Theorem afvpcfv0 27881
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 27867 . . 3  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
21eqeq1i 2415 . 2  |-  ( ( F''' A )  =  _V  <->  if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V )
3 eqcom 2410 . . . 4  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  _V  =  if ( F defAt  A ,  ( F `  A ) ,  _V ) )
4 eqif 3736 . . . 4  |-  ( _V  =  if ( F defAt 
A ,  ( F `
 A ) ,  _V )  <->  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V ) ) )
53, 4bitri 241 . . 3  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  ( ( F defAt 
A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V ) ) )
6 fveqvfvv 27859 . . . . . 6  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  (/) )
76eqcoms 2411 . . . . 5  |-  ( _V  =  ( F `  A )  ->  ( F `  A )  =  (/) )
87adantl 453 . . . 4  |-  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  -> 
( F `  A
)  =  (/) )
9 fvfundmfvn0 5725 . . . . . . 7  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 27845 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
119, 10sylibr 204 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  F defAt  A )
1211necon1bi 2614 . . . . 5  |-  ( -.  F defAt  A  ->  ( F `  A )  =  (/) )
1312adantr 452 . . . 4  |-  ( ( -.  F defAt  A  /\  _V  =  _V )  ->  ( F `  A
)  =  (/) )
148, 13jaoi 369 . . 3  |-  ( ( ( F defAt  A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V )
)  ->  ( F `  A )  =  (/) )
155, 14sylbi 188 . 2  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  ->  ( F `  A )  =  (/) )
162, 15sylbi 188 1  |-  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   _Vcvv 2920   (/)c0 3592   ifcif 3703   {csn 3778   dom cdm 4841    |` cres 4843   Fun wfun 5411   ` cfv 5417   defAt wdfat 27842  '''cafv 27843
This theorem is referenced by:  afvfv0bi  27887  aovpcov0  27925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-res 4853  df-iota 5381  df-fun 5419  df-fv 5425  df-dfat 27845  df-afv 27846
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