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Theorem afveu 46766
Description: The value of a function at a unique point, analogous to fveu 6890. (Contributed by Alexander van der Vekens, 29-Nov-2017.)
Assertion
Ref Expression
afveu (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem afveu
StepHypRef Expression
1 df-br 5154 . . . 4 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
21eubii 2574 . . 3 (∃!𝑥 𝐴𝐹𝑥 ↔ ∃!𝑥𝐴, 𝑥⟩ ∈ 𝐹)
3 eu2ndop1stv 46738 . . 3 (∃!𝑥𝐴, 𝑥⟩ ∈ 𝐹𝐴 ∈ V)
42, 3sylbi 216 . 2 (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ V)
5 euex 2566 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
6 eldmg 5905 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
75, 6syl5ibrcom 246 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
87impcom 406 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
9 dfdfat2 46741 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
10 afvfundmfveq 46751 . . . . . . . . 9 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
11 fveu 6890 . . . . . . . . 9 (∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
1210, 11sylan9eq 2786 . . . . . . . 8 ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
1312ex 411 . . . . . . 7 (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
149, 13sylbir 234 . . . . . 6 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
1514expcom 412 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})))
1615pm2.43a 54 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
1716adantl 480 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
188, 17mpd 15 . 2 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
194, 18mpancom 686 1 (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wex 1774  wcel 2099  ∃!weu 2557  {cab 2703  Vcvv 3462  cop 4639   cuni 4913   class class class wbr 5153  dom cdm 5682  cfv 6554   defAt wdfat 46729  '''cafv 46730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-br 5154  df-opab 5216  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6506  df-fun 6556  df-fv 6562  df-aiota 46698  df-dfat 46732  df-afv 46733
This theorem is referenced by: (None)
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