Theorem dfatsnafv2 47262

Description: Singleton of function value, analogous to fnsnfv 6896. (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 2737	. . . 4 ⊢ (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦)
2		dfatbrafv2b 47255	. . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
3	2	elvd 3440	. . . 4 ⊢ (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
4	1, 3	bitrid 283	. . 3 ⊢ (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
5	4	abbidv 2796	. 2 ⊢ (𝐹 defAt 𝐴 → {𝑦 ∣ 𝑦 = (𝐹''''𝐴)} = {𝑦 ∣ 𝐴𝐹𝑦})
6		df-sn 4575	. . 3 ⊢ {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)}
7	6	a1i 11	. 2 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)})
8		dfdfat2 47138	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9		imasng 6030	. . . 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 2775	1 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∃!weu 2562  {cab 2708  Vcvv 3434  {csn 4574   class class class wbr 5089  dom cdm 5614   “ cima 5617   defAt wdfat 47126  ''''cafv2 47218

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6433  df-fun 6479  df-fn 6480  df-dfat 47129  df-afv2 47219

This theorem is referenced by:  afv2co2  47267