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Theorem afv0fv0 27989
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 4339 . . 3  |-  (/)  e.  _V
2 eleq1a 2505 . . 3  |-  ( (/)  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V ) )
31, 2ax-mp 8 . 2  |-  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V )
4 afvvfveq 27988 . . 3  |-  ( ( F''' A )  e.  _V  ->  ( F''' A )  =  ( F `  A ) )
5 eqeq1 2442 . . . 4  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  <->  ( F `  A )  =  (/) ) )
65biimpd 199 . . 3  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) ) )
74, 6syl 16 . 2  |-  ( ( F''' A )  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F `
 A )  =  (/) ) )
83, 7mpcom 34 1  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   ` cfv 5454  '''cafv 27948
This theorem is referenced by:  afvfv0bi  27992  aov0ov0  28033
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-nul 3629  df-if 3740  df-fv 5462  df-afv 27951
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