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Theorem afvfv0bi 28015
Description: The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvfv0bi  |-  ( ( F `  A )  =  (/)  <->  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V ) )

Proof of Theorem afvfv0bi
StepHypRef Expression
1 ioran 476 . . . 4  |-  ( -.  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V ) 
<->  ( -.  ( F''' A )  =  (/)  /\ 
-.  ( F''' A )  =  _V ) )
2 df-ne 2448 . . . . . . 7  |-  ( ( F''' A )  =/=  _V  <->  -.  ( F''' A )  =  _V )
3 afvnufveq 28010 . . . . . . 7  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
42, 3sylbir 204 . . . . . 6  |-  ( -.  ( F''' A )  =  _V  ->  ( F''' A )  =  ( F `  A ) )
5 eqeq1 2289 . . . . . . . 8  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  <->  ( F `  A )  =  (/) ) )
65notbid 285 . . . . . . 7  |-  ( ( F''' A )  =  ( F `  A )  ->  ( -.  ( F''' A )  =  (/)  <->  -.  ( F `  A )  =  (/) ) )
76biimpd 198 . . . . . 6  |-  ( ( F''' A )  =  ( F `  A )  ->  ( -.  ( F''' A )  =  (/)  ->  -.  ( F `  A )  =  (/) ) )
84, 7syl 15 . . . . 5  |-  ( -.  ( F''' A )  =  _V  ->  ( -.  ( F''' A )  =  (/)  ->  -.  ( F `  A )  =  (/) ) )
98impcom 419 . . . 4  |-  ( ( -.  ( F''' A )  =  (/)  /\  -.  ( F''' A )  =  _V )  ->  -.  ( F `  A )  =  (/) )
101, 9sylbi 187 . . 3  |-  ( -.  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V )  ->  -.  ( F `  A )  =  (/) )
1110con4i 122 . 2  |-  ( ( F `  A )  =  (/)  ->  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V ) )
12 afv0fv0 28012 . . 3  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )
13 afvpcfv0 28009 . . 3  |-  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) )
1412, 13jaoi 368 . 2  |-  ( ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V )  ->  ( F `  A )  =  (/) )
1511, 14impbii 180 1  |-  ( ( F `  A )  =  (/)  <->  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    =/= wne 2446   _Vcvv 2788   (/)c0 3455   ` cfv 5255  '''cafv 27972
This theorem is referenced by:  aovov0bi  28056
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
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