Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afvfv0bi Structured version   Unicode version

Theorem afvfv0bi 27983
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 2600 . . . . . . 7  |-  ( ( F''' A )  =/=  _V  <->  -.  ( F''' A )  =  _V )
3 afvnufveq 27978 . . . . . . 7  |-  ( ( F''' A )  =/=  _V  ->  ( F''' A )  =  ( F `  A ) )
42, 3sylbir 205 . . . . . 6  |-  ( -.  ( F''' A )  =  _V  ->  ( F''' A )  =  ( F `  A ) )
5 eqeq1 2441 . . . . . . . 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 27980 . . 3  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )
13 afvpcfv0 27977 . . 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 1652    =/= wne 2598   _Vcvv 2948   (/)c0 3620   ` cfv 5446  '''cafv 27939
This theorem is referenced by:  aovov0bi  28027
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-res 4882  df-iota 5410  df-fun 5448  df-fv 5454  df-dfat 27941  df-afv 27942
  Copyright terms: Public domain W3C validator