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Theorem fvfundmfvn0 5754
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 5747 . . . 4  |-  ( -.  A  e.  dom  F  ->  ( F `  A
)  =  (/) )
3 nfunsn 5753 . . . 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 2640 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 1652    e. wcel 1725    =/= wne 2598   (/)c0 3620   {csn 3806   dom cdm 4870    |` cres 4872   Fun wfun 5440   ` cfv 5446
This theorem is referenced by:  usgranloopv  21390  afvpcfv0  27977  afvfvn0fveq  27981  afv0nbfvbi  27982
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-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
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