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Theorem afv0nbfvbi 27983
Description: The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afv0nbfvbi  |-  ( (/)  e/  B  ->  ( ( F''' A )  e.  B  <->  ( F `  A )  e.  B ) )

Proof of Theorem afv0nbfvbi
StepHypRef Expression
1 afvvfveq 27980 . . 3  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
2 eleq1 2496 . . . 4  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  e.  B  <->  ( F `  A )  e.  B ) )
32biimpd 199 . . 3  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  e.  B  ->  ( F `  A
)  e.  B ) )
41, 3mpcom 34 . 2  |-  ( ( F''' A )  e.  B  ->  ( F `  A
)  e.  B )
5 df-nel 2602 . . . . . . 7  |-  ( (/)  e/  B  <->  -.  (/)  e.  B
)
6 nelne2 2689 . . . . . . 7  |-  ( ( ( F `  A
)  e.  B  /\  -.  (/)  e.  B )  ->  ( F `  A )  =/=  (/) )
75, 6sylan2b 462 . . . . . 6  |-  ( ( ( F `  A
)  e.  B  /\  (/) 
e/  B )  -> 
( F `  A
)  =/=  (/) )
87ancoms 440 . . . . 5  |-  ( (
(/)  e/  B  /\  ( F `  A )  e.  B )  -> 
( F `  A
)  =/=  (/) )
9 fvfundmfvn0 5755 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 27942 . . . . . . . 8  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
11 afvfundmfveq 27970 . . . . . . . 8  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
1210, 11sylbir 205 . . . . . . 7  |-  ( ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  ->  ( F''' A )  =  ( F `  A ) )
13 eleq1 2496 . . . . . . . . 9  |-  ( ( F `  A )  =  ( F''' A )  ->  ( ( F `
 A )  e.  B  <->  ( F''' A )  e.  B ) )
1413eqcoms 2439 . . . . . . . 8  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F `
 A )  e.  B  <->  ( F''' A )  e.  B ) )
1514biimpd 199 . . . . . . 7  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F `
 A )  e.  B  ->  ( F''' A )  e.  B ) )
1612, 15syl 16 . . . . . 6  |-  ( ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  ->  ( ( F `
 A )  e.  B  ->  ( F''' A )  e.  B ) )
179, 16syl 16 . . . . 5  |-  ( ( F `  A )  =/=  (/)  ->  ( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) )
188, 17syl 16 . . . 4  |-  ( (
(/)  e/  B  /\  ( F `  A )  e.  B )  -> 
( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) )
1918ex 424 . . 3  |-  ( (/)  e/  B  ->  ( ( F `  A )  e.  B  ->  ( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) ) )
2019pm2.43d 46 . 2  |-  ( (/)  e/  B  ->  ( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) )
214, 20impbid2 196 1  |-  ( (/)  e/  B  ->  ( ( F''' A )  e.  B  <->  ( F `  A )  e.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    e/ wnel 2600   (/)c0 3621   {csn 3807   dom cdm 4871    |` cres 4873   Fun wfun 5441   ` cfv 5447   defAt wdfat 27939  '''cafv 27940
This theorem is referenced by:  aov0nbovbi  28027
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 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-res 4883  df-iota 5411  df-fun 5449  df-fv 5455  df-dfat 27942  df-afv 27943
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