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Theorem fvfundmfvn0 5913
Description: If a class' 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 490 . . 3  |-  ( -.  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  <->  ( -.  A  e.  dom  F  \/  -.  Fun  ( F  |`  { A } ) ) )
2 ndmfv 5905 . . . 4  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
3 nfunsn 5912 . . . 4  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
42, 3jaoi 380 . . 3  |-  ( ( -.  A  e.  dom  F  \/  -.  Fun  ( F  |`  { A }
) )  ->  ( F `  A )  =  (/) )
51, 4sylbi 198 . 2  |-  ( -.  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  ->  ( F `  A )  =  (/) )
65necon1ai 2662 1  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   (/)c0 3767   {csn 4002   dom cdm 4854    |` cres 4856   Fun wfun 5595   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609
This theorem is referenced by:  fvn0ssdmfun  6028  fvn0fvelrn  6096  usgranloopv  24951  afvpcfv0  38047  afvfvn0fveq  38051  afv0nbfvbi  38052  ovn0dmfun  38531
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