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Theorem afv0nbfvbi 32475
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 32472 . . 3  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
2 eleq1 2526 . . . 4  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  e.  B  <->  ( F `  A )  e.  B ) )
32biimpd 207 . . 3  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  e.  B  ->  ( F `  A
)  e.  B ) )
41, 3mpcom 36 . 2  |-  ( ( F''' A )  e.  B  ->  ( F `  A
)  e.  B )
5 elnelne2 2802 . . . . . 6  |-  ( ( ( F `  A
)  e.  B  /\  (/) 
e/  B )  -> 
( F `  A
)  =/=  (/) )
65ancoms 451 . . . . 5  |-  ( (
(/)  e/  B  /\  ( F `  A )  e.  B )  -> 
( F `  A
)  =/=  (/) )
7 fvfundmfvn0 5880 . . . . 5  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
8 df-dfat 32440 . . . . . 6  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
9 afvfundmfveq 32462 . . . . . 6  |-  ( F defAt 
A  ->  ( F''' A )  =  ( F `
 A ) )
108, 9sylbir 213 . . . . 5  |-  ( ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) )  ->  ( F''' A )  =  ( F `  A ) )
11 eleq1 2526 . . . . . . 7  |-  ( ( F `  A )  =  ( F''' A )  ->  ( ( F `
 A )  e.  B  <->  ( F''' A )  e.  B ) )
1211eqcoms 2466 . . . . . 6  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F `
 A )  e.  B  <->  ( F''' A )  e.  B ) )
1312biimpd 207 . . . . 5  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F `
 A )  e.  B  ->  ( F''' A )  e.  B ) )
146, 7, 10, 134syl 21 . . . 4  |-  ( (
(/)  e/  B  /\  ( F `  A )  e.  B )  -> 
( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) )
1514ex 432 . . 3  |-  ( (/)  e/  B  ->  ( ( F `  A )  e.  B  ->  ( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) ) )
1615pm2.43d 48 . 2  |-  ( (/)  e/  B  ->  ( ( F `  A )  e.  B  ->  ( F''' A )  e.  B
) )
174, 16impbid2 204 1  |-  ( (/)  e/  B  ->  ( ( F''' A )  e.  B  <->  ( F `  A )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    e/ wnel 2650   (/)c0 3783   {csn 4016   dom cdm 4988    |` cres 4990   Fun wfun 5564   ` cfv 5570   defAt wdfat 32437  '''cafv 32438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-res 5000  df-iota 5534  df-fun 5572  df-fv 5578  df-dfat 32440  df-afv 32441
This theorem is referenced by:  aov0nbovbi  32519
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