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Theorem fvfundmfvn0 5720
Description: If a class's value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
fvfundmfvn0  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )

Proof of Theorem fvfundmfvn0
StepHypRef Expression
1 ianor 488 . . 3  |-  ( -.  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  <->  ( -.  A  e.  dom  F  \/  -.  Fun  ( F  |`  { A } ) ) )
2 ndmfv 5712 . . . 4  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
3 nfunsn 5719 . . . 4  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
42, 3jaoi 379 . . 3  |-  ( ( -.  A  e.  dom  F  \/  -.  Fun  ( F  |`  { A }
) )  ->  ( F `  A )  =  (/) )
51, 4sylbi 195 . 2  |-  ( -.  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  ->  ( F `  A )  =  (/) )
65necon1ai 2651 1  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   (/)c0 3635   {csn 3875   dom cdm 4838    |` cres 4840   Fun wfun 5410   ` cfv 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-res 4850  df-iota 5379  df-fun 5418  df-fv 5424
This theorem is referenced by:  usgranloopv  23295  afvpcfv0  30049  afvfvn0fveq  30053  afv0nbfvbi  30054  fvn0fvelrn  30150
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