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Theorem afv0fv0 28117
Description: If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afv0fv0  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )

Proof of Theorem afv0fv0
StepHypRef Expression
1 0ex 4166 . . 3  |-  (/)  e.  _V
2 eleq1a 2365 . . 3  |-  ( (/)  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V ) )
31, 2ax-mp 8 . 2  |-  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V )
4 afvvfveq 28116 . . 3  |-  ( ( F''' A )  e.  _V  ->  ( F''' A )  =  ( F `  A ) )
5 eqeq1 2302 . . . 4  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  <->  ( F `  A )  =  (/) ) )
65biimpd 198 . . 3  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) ) )
74, 6syl 15 . 2  |-  ( ( F''' A )  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F `
 A )  =  (/) ) )
83, 7mpcom 32 1  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   ` cfv 5271  '''cafv 28075
This theorem is referenced by:  afvfv0bi  28120  aov0ov0  28161
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-if 3579  df-fv 5279  df-afv 28078
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