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Theorem dfafv23 45559
Description: A definition of function value in terms of iota, analogous to dffv3 6843. (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 45516 . 2 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2 dfdfat2 45434 . . . . . . 7 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
32simplbi 499 . . . . . 6 (𝐹 defAt 𝐴𝐴 ∈ dom 𝐹)
4 elimasng 6045 . . . . . 6 ((𝐴 ∈ dom 𝐹𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
53, 4sylan 581 . . . . 5 ((𝐹 defAt 𝐴𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
6 df-br 5111 . . . . 5 (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
75, 6bitr4di 289 . . . 4 ((𝐹 defAt 𝐴𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
87elvd 3455 . . 3 (𝐹 defAt 𝐴 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
98iotabidv 6485 . 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 205  wa 397   = wceq 1542  wcel 2107  ∃!weu 2567  Vcvv 3448  {csn 4591  cop 4597   class class class wbr 5110  dom cdm 5638  cima 5641  cio 6451   defAt wdfat 45422  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-dfat 45425  df-afv2 45515
This theorem is referenced by:  afv2co2  45563
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