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Theorem dfatsnafv2 43599
Description: Singleton of function value, analogous to fnsnfv 6716. (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 2828 . . . 4 (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦)
2 dfatbrafv2b 43592 . . . . 5 ((𝐹 defAt 𝐴𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
32elvd 3477 . . . 4 (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
41, 3syl5bb 286 . . 3 (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
54abbidv 2885 . 2 (𝐹 defAt 𝐴 → {𝑦𝑦 = (𝐹''''𝐴)} = {𝑦𝐴𝐹𝑦})
6 df-sn 4541 . . 3 {(𝐹''''𝐴)} = {𝑦𝑦 = (𝐹''''𝐴)}
76a1i 11 . 2 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦𝑦 = (𝐹''''𝐴)})
8 dfdfat2 43475 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9 imasng 5924 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
109adantr 484 . . 3 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
118, 10sylbi 220 . 2 (𝐹 defAt 𝐴 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
125, 7, 113eqtr4d 2866 1 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  ∃!weu 2653  {cab 2799  Vcvv 3471  {csn 4540   class class class wbr 5039  dom cdm 5528  cima 5531   defAt wdfat 43463  ''''cafv2 43555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-dfat 43466  df-afv2 43556
This theorem is referenced by:  afv2co2  43604
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