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Theorem fvfundmfvn0 27656
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 475 . . 3  |-  ( -.  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  <->  ( -.  A  e.  dom  F  \/  -.  Fun  ( F  |`  { A } ) ) )
2 ndmfv 5695 . . . 4  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
3 nfunsn 5701 . . . 4  |-  ( -. 
Fun  ( F  |`  { A } )  -> 
( F `  A
)  =  (/) )
42, 3jaoi 369 . . 3  |-  ( ( -.  A  e.  dom  F  \/  -.  Fun  ( F  |`  { A }
) )  ->  ( F `  A )  =  (/) )
51, 4sylbi 188 . 2  |-  ( -.  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  ->  ( F `  A )  =  (/) )
65necon1ai 2592 1  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   (/)c0 3571   {csn 3757   dom cdm 4818    |` cres 4820   Fun wfun 5388   ` cfv 5394
This theorem is referenced by:  afvpcfv0  27679  afvfvn0fveq  27683  afv0nbfvbi  27684
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 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-iota 5358  df-fun 5396  df-fv 5402
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