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Theorem afv0fv0 38369
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 4554 . . 3  |-  (/)  e.  _V
2 eleq1a 2506 . . 3  |-  ( (/)  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V ) )
31, 2ax-mp 5 . 2  |-  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V )
4 afvvfveq 38368 . . 3  |-  ( ( F''' A )  e.  _V  ->  ( F''' A )  =  ( F `  A ) )
5 eqeq1 2427 . . . 4  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  <->  ( F `  A )  =  (/) ) )
65biimpd 211 . . 3  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) ) )
74, 6syl 17 . 2  |-  ( ( F''' A )  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F `
 A )  =  (/) ) )
83, 7mpcom 38 1  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    e. wcel 1869   _Vcvv 3082   (/)c0 3762   ` cfv 5599  '''cafv 38334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-nul 3763  df-if 3911  df-fv 5607  df-afv 38337
This theorem is referenced by:  afvfv0bi  38372  aov0ov0  38413
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