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Theorem afv0nbfvbi 27684
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 27681 . . 3  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
2 eleq1 2447 . . . 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 2553 . . . . . . 7  |-  ( (/)  e/  B  <->  -.  (/)  e.  B
)
6 nelne2 2640 . . . . . . 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 27656 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 27642 . . . . . . . 8  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
11 afvfundmfveq 27671 . . . . . . . 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 2447 . . . . . . . . 9  |-  ( ( F `  A )  =  ( F''' A )  ->  ( ( F `
 A )  e.  B  <->  ( F''' A )  e.  B ) )
1413eqcoms 2390 . . . . . . . 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 1649    e. wcel 1717    =/= wne 2550    e/ wnel 2551   (/)c0 3571   {csn 3757   dom cdm 4818    |` cres 4820   Fun wfun 5388   ` cfv 5394   defAt wdfat 27639  '''cafv 27640
This theorem is referenced by:  aov0nbovbi  27728
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-nel 2553  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  df-dfat 27642  df-afv 27643
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