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Theorem fvfundmfvn0 5898
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 5890 . . . 4  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
3 nfunsn 5897 . . . 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 2698 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 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   {csn 4027   dom cdm 4999    |` cres 5001   Fun wfun 5582   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596
This theorem is referenced by:  fvn0fvelrn  6078  usgranloopv  24082  afvpcfv0  31726  afvfvn0fveq  31730  afv0nbfvbi  31731
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