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Theorem afv0fv0 28012
 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 4150 . . 3
2 eleq1a 2352 . . 3 ''' '''
31, 2ax-mp 8 . 2 ''' '''
4 afvvfveq 28011 . . 3 ''' '''
5 eqeq1 2289 . . . 4 ''' '''
65biimpd 198 . . 3 ''' '''
74, 6syl 15 . 2 ''' '''
83, 7mpcom 32 1 '''
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1623   wcel 1684  cvv 2788  c0 3455  cfv 5255  '''cafv 27972 This theorem is referenced by:  afvfv0bi  28015  aov0ov0  28053 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-afv 27975
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