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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  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )

Proof of Theorem afv0fv0
StepHypRef Expression
1 0ex 4150 . . 3  |-  (/)  e.  _V
2 eleq1a 2352 . . 3  |-  ( (/)  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V ) )
31, 2ax-mp 8 . 2  |-  ( ( F''' A )  =  (/)  ->  ( F''' A )  e.  _V )
4 afvvfveq 28011 . . 3  |-  ( ( F''' A )  e.  _V  ->  ( F''' A )  =  ( F `  A ) )
5 eqeq1 2289 . . . 4  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  <->  ( F `  A )  =  (/) ) )
65biimpd 198 . . 3  |-  ( ( F''' A )  =  ( F `  A )  ->  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) ) )
74, 6syl 15 . 2  |-  ( ( F''' A )  e.  _V  ->  ( ( F''' A )  =  (/)  ->  ( F `
 A )  =  (/) ) )
83, 7mpcom 32 1  |-  ( ( F''' A )  =  (/)  ->  ( F `  A
)  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 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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