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Theorem afvfv0bi 27678
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 477 . . . 4  |-  ( -.  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V ) 
<->  ( -.  ( F''' A )  =  (/)  /\ 
-.  ( F''' A )  =  _V ) )
2 df-ne 2545 . . . . . . 7  |-  ( ( F''' A )  =/=  _V  <->  -.  ( F''' A )  =  _V )
3 afvnufveq 27673 . . . . . . 7  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
42, 3sylbir 205 . . . . . 6  |-  ( -.  ( F''' A )  =  _V  ->  ( F''' A )  =  ( F `  A ) )
5 eqeq1 2386 . . . . . . . 8  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  <->  ( F `  A )  =  (/) ) )
65notbid 286 . . . . . . 7  |-  ( ( F''' A )  =  ( F `  A )  ->  ( -.  ( F''' A )  =  (/)  <->  -.  ( F `  A )  =  (/) ) )
76biimpd 199 . . . . . 6  |-  ( ( F''' A )  =  ( F `  A )  ->  ( -.  ( F''' A )  =  (/)  ->  -.  ( F `  A )  =  (/) ) )
84, 7syl 16 . . . . 5  |-  ( -.  ( F''' A )  =  _V  ->  ( -.  ( F''' A )  =  (/)  ->  -.  ( F `  A )  =  (/) ) )
98impcom 420 . . . 4  |-  ( ( -.  ( F''' A )  =  (/)  /\  -.  ( F''' A )  =  _V )  ->  -.  ( F `  A )  =  (/) )
101, 9sylbi 188 . . 3  |-  ( -.  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V )  ->  -.  ( F `  A )  =  (/) )
1110con4i 124 . 2  |-  ( ( F `  A )  =  (/)  ->  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V ) )
12 afv0fv0 27675 . . 3  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )
13 afvpcfv0 27672 . . 3  |-  ( ( F''' A )  =  _V  ->  ( F `  A
)  =  (/) )
1412, 13jaoi 369 . 2  |-  ( ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V )  ->  ( F `  A )  =  (/) )
1511, 14impbii 181 1  |-  ( ( F `  A )  =  (/)  <->  ( ( F''' A )  =  (/)  \/  ( F''' A )  =  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    =/= wne 2543   _Vcvv 2892   (/)c0 3564   ` cfv 5387  '''cafv 27633
This theorem is referenced by:  aovov0bi  27722
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-res 4823  df-iota 5351  df-fun 5389  df-fv 5395  df-dfat 27635  df-afv 27636
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