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Theorem afveu 41731
Description: The value of a function at a unique point, analogous to fveu 6336. (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 4797 . . . 4 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
21eubii 2621 . . 3 (∃!𝑥 𝐴𝐹𝑥 ↔ ∃!𝑥𝐴, 𝑥⟩ ∈ 𝐹)
3 eu2ndop1stv 41700 . . 3 (∃!𝑥𝐴, 𝑥⟩ ∈ 𝐹𝐴 ∈ V)
42, 3sylbi 207 . 2 (∃!𝑥 𝐴𝐹𝑥𝐴 ∈ V)
5 euex 2623 . . . . 5 (∃!𝑥 𝐴𝐹𝑥 → ∃𝑥 𝐴𝐹𝑥)
6 eldmg 5466 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ dom 𝐹 ↔ ∃𝑥 𝐴𝐹𝑥))
75, 6syl5ibrcom 237 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ V → 𝐴 ∈ dom 𝐹))
87impcom 445 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → 𝐴 ∈ dom 𝐹)
9 dfdfat2 41709 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
10 afvfundmfveq 41716 . . . . . . . . 9 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
11 fveu 6336 . . . . . . . . 9 (∃!𝑥 𝐴𝐹𝑥 → (𝐹𝐴) = {𝑥𝐴𝐹𝑥})
1210, 11sylan9eq 2806 . . . . . . . 8 ((𝐹 defAt 𝐴 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
1312ex 449 . . . . . . 7 (𝐹 defAt 𝐴 → (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
149, 13sylbir 225 . . . . . 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 54 . . . 4 (∃!𝑥 𝐴𝐹𝑥 → (𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
1716adantl 473 . . 3 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐴 ∈ dom 𝐹 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥}))
188, 17mpd 15 . 2 ((𝐴 ∈ V ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
194, 18mpancom 706 1 (∃!𝑥 𝐴𝐹𝑥 → (𝐹'''𝐴) = {𝑥𝐴𝐹𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1624  wex 1845  wcel 2131  ∃!weu 2599  {cab 2738  Vcvv 3332  cop 4319   cuni 4580   class class class wbr 4796  dom cdm 5258  cfv 6041   defAt wdfat 41691  '''cafv 41692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-res 5270  df-iota 6004  df-fun 6043  df-fv 6049  df-dfat 41694  df-afv 41695
This theorem is referenced by: (None)
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