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Theorem dfatsnafv2 45558
Description: Singleton of function value, analogous to fnsnfv 6925. (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 2744 . . . 4 (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦)
2 dfatbrafv2b 45551 . . . . 5 ((𝐹 defAt 𝐴𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
32elvd 3455 . . . 4 (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
41, 3bitrid 283 . . 3 (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
54abbidv 2806 . 2 (𝐹 defAt 𝐴 → {𝑦𝑦 = (𝐹''''𝐴)} = {𝑦𝐴𝐹𝑦})
6 df-sn 4592 . . 3 {(𝐹''''𝐴)} = {𝑦𝑦 = (𝐹''''𝐴)}
76a1i 11 . 2 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦𝑦 = (𝐹''''𝐴)})
8 dfdfat2 45434 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9 imasng 6040 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
109adantr 482 . . 3 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
118, 10sylbi 216 . 2 (𝐹 defAt 𝐴 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
125, 7, 113eqtr4d 2787 1 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  ∃!weu 2567  {cab 2714  Vcvv 3448  {csn 4591   class class class wbr 5110  dom cdm 5638  cima 5641   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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  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-fn 6504  df-dfat 45425  df-afv2 45515
This theorem is referenced by:  afv2co2  45563
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