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Theorem afveu 39882
Description: The value of a function at a unique point, analogous to fveu 6095. (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 4584 . . . 4 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
21eubii 2480 . . 3 (∃!𝑥 𝐴𝐹𝑥 ↔ ∃!𝑥𝐴, 𝑥⟩ ∈ 𝐹)
3 eu2ndop1stv 39851 . . 3 (∃!𝑥𝐴, 𝑥⟩ ∈ 𝐹𝐴 ∈ V)
42, 3sylbi 206 . 2 (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ V)
5 euex 2482 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
6 eldmg 5241 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
75, 6syl5ibrcom 236 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
87impcom 445 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
9 dfdfat2 39860 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
10 afvfundmfveq 39867 . . . . . . . . 9 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
11 fveu 6095 . . . . . . . . 9 (∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
1210, 11sylan9eq 2664 . . . . . . . 8 ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
1312ex 449 . . . . . . 7 (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
149, 13sylbir 224 . . . . . 6 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
1514expcom 450 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})))
1615pm2.43a 52 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
1716adantl 481 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
188, 17mpd 15 . 2 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
194, 18mpancom 700 1 (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wex 1695  wcel 1977  ∃!weu 2458  {cab 2596  Vcvv 3173  cop 4131   cuni 4372   class class class wbr 4583  dom cdm 5038  cfv 5804   defAt wdfat 39842  '''cafv 39843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-dfat 39845  df-afv 39846
This theorem is referenced by: (None)
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