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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 '''

Proof of Theorem afv0fv0
StepHypRef Expression
1 0ex 4554 . . 3
2 eleq1a 2506 . . 3 ''' '''
31, 2ax-mp 5 . 2 ''' '''
4 afvvfveq 38368 . . 3 ''' '''
5 eqeq1 2427 . . . 4 ''' '''
65biimpd 211 . . 3 ''' '''
74, 6syl 17 . 2 ''' '''
83, 7mpcom 38 1 '''
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1438   wcel 1869  cvv 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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