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Theorem afvpcfv0 40560
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 40546 . . 3 (𝐹'''𝐴) = if(𝐹 defAt 𝐴, (𝐹𝐴), V)
21eqeq1i 2626 . 2 ((𝐹'''𝐴) = V ↔ if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V)
3 eqcom 2628 . . . 4 (if(𝐹 defAt 𝐴, (𝐹𝐴), V) = V ↔ V = if(𝐹 defAt 𝐴, (𝐹𝐴), V))
4 eqif 4104 . . . 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 40538 . . . . . 6 ((𝐹𝐴) = V → (𝐹𝐴) = ∅)
76eqcoms 2629 . . . . 5 (V = (𝐹𝐴) → (𝐹𝐴) = ∅)
87adantl 482 . . . 4 ((𝐹 defAt 𝐴 ∧ V = (𝐹𝐴)) → (𝐹𝐴) = ∅)
9 fvfundmfvn0 6193 . . . . . . 7 ((𝐹𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
10 df-dfat 40530 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
119, 10sylibr 224 . . . . . 6 ((𝐹𝐴) ≠ ∅ → 𝐹 defAt 𝐴)
1211necon1bi 2818 . . . . 5 𝐹 defAt 𝐴 → (𝐹𝐴) = ∅)
1312adantr 481 . . . 4 ((¬ 𝐹 defAt 𝐴 ∧ V = V) → (𝐹𝐴) = ∅)
148, 13jaoi 394 . . 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  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3190  c0 3897  ifcif 4064  {csn 4155  dom cdm 5084  cres 5086  Fun wfun 5851  cfv 5857   defAt wdfat 40527  '''cafv 40528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-res 5096  df-iota 5820  df-fun 5859  df-fv 5865  df-dfat 40530  df-afv 40531
This theorem is referenced by:  afvfv0bi  40566  aovpcov0  40604
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