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Theorem dfatsnafv2 47269
Description: Singleton of function value, analogous to fnsnfv 6987. (Contributed by AV, 7-Sep-2022.)
Assertion
Ref Expression
dfatsnafv2 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))

Proof of Theorem dfatsnafv2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2743 . . . 4 (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦)
2 dfatbrafv2b 47262 . . . . 5 ((𝐹 defAt 𝐴𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
32elvd 3485 . . . 4 (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
41, 3bitrid 283 . . 3 (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
54abbidv 2807 . 2 (𝐹 defAt 𝐴 → {𝑦𝑦 = (𝐹''''𝐴)} = {𝑦𝐴𝐹𝑦})
6 df-sn 4626 . . 3 {(𝐹''''𝐴)} = {𝑦𝑦 = (𝐹''''𝐴)}
76a1i 11 . 2 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦𝑦 = (𝐹''''𝐴)})
8 dfdfat2 47145 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9 imasng 6101 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
109adantr 480 . . 3 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
118, 10sylbi 217 . 2 (𝐹 defAt 𝐴 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
125, 7, 113eqtr4d 2786 1 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  ∃!weu 2567  {cab 2713  Vcvv 3479  {csn 4625   class class class wbr 5142  dom cdm 5684  cima 5687   defAt wdfat 47133  ''''cafv2 47225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-dfat 47136  df-afv2 47226
This theorem is referenced by:  afv2co2  47274
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