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Theorem afvpcfv0 43493
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 43479 . . 3 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
21eqeq1i 2825 . 2 ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V)
3 eqcom 2827 . . . 4 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹𝐴), V))
4 eqif 4483 . . . 4 (V = if(𝐹 defAt 𝐴, (𝐹𝐴), V) ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)))
53, 4bitri 277 . . 3 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V ↔ ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)))
6 fveqvfvv 43423 . . . . . 6 ((𝐹𝐴) = V → (𝐹𝐴) = ∅)
76eqcoms 2828 . . . . 5 (V = (𝐹𝐴) → (𝐹𝐴) = ∅)
87adantl 484 . . . 4 ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) → (𝐹𝐴) = ∅)
9 fvfundmfvn0 6684 . . . . . . 7 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 43466 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
119, 10sylibr 236 . . . . . 6 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
1211necon1bi 3034 . . . . 5 𝐹 defAt 𝐴 → (𝐹𝐴) = ∅)
1312adantr 483 . . . 4 ((¬ 𝐹 defAt 𝐴 ∧ V = V) → (𝐹𝐴) = ∅)
148, 13jaoi 853 . . 3 (((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) ∨ (¬ 𝐹 defAt 𝐴 ∧ V = V)) → (𝐹𝐴) = ∅)
155, 14sylbi 219 . 2 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V → (𝐹𝐴) = ∅)
162, 15sylbi 219 1 ((𝐹'''𝐴) = V → (𝐹𝐴) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843   = wceq 1537  wcel 2114  wne 3006  Vcvv 3473  c0 4269  ifcif 4443  {csn 4543  dom cdm 5531  cres 5533  Fun wfun 6325  cfv 6331   defAt wdfat 43463  '''cafv 43464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-int 4853  df-br 5043  df-opab 5105  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-res 5543  df-iota 6290  df-fun 6333  df-fv 6339  df-aiota 43433  df-dfat 43466  df-afv 43467
This theorem is referenced by:  afvfv0bi  43499  aovpcov0  43537
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