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Theorem dfafv23 47265
Description: A definition of function value in terms of iota, analogous to dffv3 6902. (Contributed by AV, 6-Sep-2022.)
Assertion
Ref Expression
dfafv23 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem dfafv23
StepHypRef Expression
1 dfatafv2iota 47222 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2 dfdfat2 47140 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
32simplbi 497 . . . . . 6 (𝐹 defAt 𝐴𝐴 ∈ dom 𝐹)
4 elimasng 6107 . . . . . 6 ((𝐴 ∈ dom 𝐹𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
53, 4sylan 580 . . . . 5 ((𝐹 defAt 𝐴𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
6 df-br 5144 . . . . 5 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
75, 6bitr4di 289 . . . 4 ((𝐹 defAt 𝐴𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
87elvd 3486 . . 3 (𝐹 defAt 𝐴 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
98iotabidv 6545 . 2 (𝐹 defAt 𝐴 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
101, 9eqtr4d 2780 1 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  ∃!weu 2568  Vcvv 3480  {csn 4626  cop 4632   class class class wbr 5143  dom cdm 5685  cima 5688  cio 6512   defAt wdfat 47128  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-dfat 47131  df-afv2 47221
This theorem is referenced by:  afv2co2  47269
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