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Theorem afvpcfv0 32185
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 32171 . . 3  |-  ( F''' A )  =  if ( F defAt  A , 
( F `  A
) ,  _V )
21eqeq1i 2450 . 2  |-  ( ( F''' A )  =  _V  <->  if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V )
3 eqcom 2452 . . . 4  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  _V  =  if ( F defAt  A ,  ( F `  A ) ,  _V ) )
4 eqif 3964 . . . 4  |-  ( _V  =  if ( F defAt 
A ,  ( F `
 A ) ,  _V )  <->  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V ) ) )
53, 4bitri 249 . . 3  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  <->  ( ( F defAt 
A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V ) ) )
6 fveqvfvv 32163 . . . . . 6  |-  ( ( F `  A )  =  _V  ->  ( F `  A )  =  (/) )
76eqcoms 2455 . . . . 5  |-  ( _V  =  ( F `  A )  ->  ( F `  A )  =  (/) )
87adantl 466 . . . 4  |-  ( ( F defAt  A  /\  _V  =  ( F `  A ) )  -> 
( F `  A
)  =  (/) )
9 fvfundmfvn0 5888 . . . . . . 7  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 32155 . . . . . . 7  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
119, 10sylibr 212 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  F defAt  A )
1211necon1bi 2676 . . . . 5  |-  ( -.  F defAt  A  ->  ( F `  A )  =  (/) )
1312adantr 465 . . . 4  |-  ( ( -.  F defAt  A  /\  _V  =  _V )  ->  ( F `  A
)  =  (/) )
148, 13jaoi 379 . . 3  |-  ( ( ( F defAt  A  /\  _V  =  ( F `  A ) )  \/  ( -.  F defAt  A  /\  _V  =  _V )
)  ->  ( F `  A )  =  (/) )
155, 14sylbi 195 . 2  |-  ( if ( F defAt  A , 
( F `  A
) ,  _V )  =  _V  ->  ( F `  A )  =  (/) )
162, 15sylbi 195 1  |-  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095   (/)c0 3770   ifcif 3926   {csn 4014   dom cdm 4989    |` cres 4991   Fun wfun 5572   ` cfv 5578   defAt wdfat 32152  '''cafv 32153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-res 5001  df-iota 5541  df-fun 5580  df-fv 5586  df-dfat 32155  df-afv 32156
This theorem is referenced by:  afvfv0bi  32191  aovpcov0  32229
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