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Theorem afv0nbfvbi 28119
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 28116 . . 3  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
2 eleq1 2356 . . . 4  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  e.  B  <->  ( F `  A )  e.  B ) )
32biimpd 198 . . 3  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  e.  B  ->  ( F `  A
)  e.  B ) )
41, 3mpcom 32 . 2  |-  ( ( F''' A )  e.  B  ->  ( F `  A
)  e.  B )
5 df-nel 2462 . . . . . . 7  |-  ( (/)  e/  B  <->  -.  (/)  e.  B
)
6 nelne2 2549 . . . . . . 7  |-  ( ( ( F `  A
)  e.  B  /\  -.  (/)  e.  B )  ->  ( F `  A )  =/=  (/) )
75, 6sylan2b 461 . . . . . 6  |-  ( ( ( F `  A
)  e.  B  /\  (/) 
e/  B )  -> 
( F `  A
)  =/=  (/) )
87ancoms 439 . . . . 5  |-  ( (
(/)  e/  B  /\  ( F `  A )  e.  B )  -> 
( F `  A
)  =/=  (/) )
9 fvfundmfvn0 28091 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 28077 . . . . . . . 8  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
11 afvfundmfveq 28106 . . . . . . . 8  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
1210, 11sylbir 204 . . . . . . 7  |-  ( ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  ->  ( F''' A )  =  ( F `  A ) )
13 eleq1 2356 . . . . . . . . 9  |-  ( ( F `  A )  =  ( F''' A )  ->  ( ( F `
 A )  e.  B  <->  ( F''' A )  e.  B ) )
1413eqcoms 2299 . . . . . . . 8  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F `
 A )  e.  B  <->  ( F''' A )  e.  B ) )
1514biimpd 198 . . . . . . 7  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F `
 A )  e.  B  ->  ( F''' A )  e.  B ) )
1612, 15syl 15 . . . . . 6  |-  ( ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  ->  ( ( F `
 A )  e.  B  ->  ( F''' A )  e.  B ) )
179, 16syl 15 . . . . 5  |-  ( ( F `  A )  =/=  (/)  ->  ( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) )
188, 17syl 15 . . . 4  |-  ( (
(/)  e/  B  /\  ( F `  A )  e.  B )  -> 
( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) )
1918ex 423 . . 3  |-  ( (/)  e/  B  ->  ( ( F `  A )  e.  B  ->  ( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) ) )
2019pm2.43d 44 . 2  |-  ( (/)  e/  B  ->  ( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) )
214, 20impbid2 195 1  |-  ( (/)  e/  B  ->  ( ( F''' A )  e.  B  <->  ( F `  A )  e.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    e/ wnel 2460   (/)c0 3468   {csn 3653   dom cdm 4705    |` cres 4707   Fun wfun 5265   ` cfv 5271   defAt wdfat 28074  '''cafv 28075
This theorem is referenced by:  aov0nbovbi  28163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fv 5279  df-dfat 28077  df-afv 28078
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