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Theorem afv0fv0 32473
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 4569 . . 3  |-  (/)  e.  _V
2 eleq1a 2537 . . 3  |-  ( (/)  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V ) )
31, 2ax-mp 5 . 2  |-  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V )
4 afvvfveq 32472 . . 3  |-  ( ( F''' A )  e.  _V  ->  ( F''' A )  =  ( F `  A ) )
5 eqeq1 2458 . . . 4  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  <->  ( F `  A )  =  (/) ) )
65biimpd 207 . . 3  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) ) )
74, 6syl 16 . 2  |-  ( ( F''' A )  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F `
 A )  =  (/) ) )
83, 7mpcom 36 1  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   ` cfv 5570  '''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
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-nul 3784  df-if 3930  df-fv 5578  df-afv 32441
This theorem is referenced by:  afvfv0bi  32476  aov0ov0  32517
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