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Theorem afv0nbfvbi 28014
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 28011 . . 3  |-  ( ( F''' A )  e.  B  ->  ( F''' A )  =  ( F `  A ) )
2 eleq1 2343 . . . 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 2449 . . . . . . 7  |-  ( (/)  e/  B  <->  -.  (/)  e.  B
)
6 nelne2 2536 . . . . . . 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 27986 . . . . . 6  |-  ( ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
10 df-dfat 27974 . . . . . . . 8  |-  ( F defAt 
A  <->  ( A  e. 
dom  F  /\  Fun  ( F  |`  { A }
) ) )
11 afvfundmfveq 28001 . . . . . . . 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 2343 . . . . . . . . 9  |-  ( ( F `  A )  =  ( F''' A )  ->  ( ( F `
 A )  e.  B  <->  ( F''' A )  e.  B ) )
1413eqcoms 2286 . . . . . . . 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 1623    e. wcel 1684    =/= wne 2446    e/ wnel 2447   (/)c0 3455   {csn 3640   dom cdm 4689    |` cres 4691   Fun wfun 5249   ` cfv 5255   defAt wdfat 27971  '''cafv 27972
This theorem is referenced by:  aov0nbovbi  28055
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-dfat 27974  df-afv 27975
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