Theorem dfatsnafv2 47366

Description: Singleton of function value, analogous to fnsnfv 6910. (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.

Step	Hyp	Ref	Expression
1		eqcom 2740	. . . 4 ⊢ (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦)
2		dfatbrafv2b 47359	. . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
3	2	elvd 3444	. . . 4 ⊢ (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
4	1, 3	bitrid 283	. . 3 ⊢ (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
5	4	abbidv 2799	. 2 ⊢ (𝐹 defAt 𝐴 → {𝑦 ∣ 𝑦 = (𝐹''''𝐴)} = {𝑦 ∣ 𝐴𝐹𝑦})
6		df-sn 4578	. . 3 ⊢ {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)}
7	6	a1i 11	. 2 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)})
8		dfdfat2 47242	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9		imasng 6040	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
10	9	adantr 480	. . 3 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
11	8, 10	sylbi 217	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
12	5, 7, 11	3eqtr4d 2778	1 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃!weu 2565  {cab 2711  Vcvv 3438  {csn 4577   class class class wbr 5095  dom cdm 5621   “ cima 5624   defAt wdfat 47230  ''''cafv2 47322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-dfat 47233  df-afv2 47323

This theorem is referenced by:  afv2co2  47371