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Theorem afvpcfv0 41740
Description: If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvpcfv0 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)

Proof of Theorem afvpcfv0
StepHypRef Expression
1 dfafv2 41726 . . 3 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
21eqeq1i 2775 . 2 ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V)
3 eqcom 2777 . . . 4 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹𝐴), V))
4 eqif 4263 . . . 4 (V = if(𝐹 defAt 𝐴, (𝐹𝐴), V) ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)))
53, 4bitri 264 . . 3 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)))
6 fveqvfvv 41718 . . . . . 6 ((𝐹𝐴) = V → (𝐹𝐴) = ∅)
76eqcoms 2778 . . . . 5 (V = (𝐹𝐴) → (𝐹𝐴) = ∅)
87adantl 467 . . . 4 ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) → (𝐹𝐴) = ∅)
9 fvfundmfvn0 6367 . . . . . . 7 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 41710 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
119, 10sylibr 224 . . . . . 6 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
1211necon1bi 2970 . . . . 5 𝐹 defAt 𝐴 → (𝐹𝐴) = ∅)
1312adantr 466 . . . 4 ((¬ 𝐹 defAt 𝐴 ∧ V = V) → (𝐹𝐴) = ∅)
148, 13jaoi 837 . . 3 (((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)) → (𝐹𝐴) = ∅)
155, 14sylbi 207 . 2 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V → (𝐹𝐴) = ∅)
162, 15sylbi 207 1 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 826   = wceq 1630  wcel 2144  wne 2942  Vcvv 3349  c0 4061  ifcif 4223  {csn 4314  dom cdm 5249  cres 5251  Fun wfun 6025  cfv 6031   defAt wdfat 41707  '''cafv 41708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-dfat 41710  df-afv 41711
This theorem is referenced by:  afvfv0bi  41746  aovpcov0  41784
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