Theorem dfatsnafv2 46695

Description: Singleton of function value, analogous to fnsnfv 6972. (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 2732	. . . 4 ⊢ (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦)
2		dfatbrafv2b 46688	. . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
3	2	elvd 3470	. . . 4 ⊢ (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦 ↔ 𝐴𝐹𝑦))
4	1, 3	bitrid 282	. . 3 ⊢ (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
5	4	abbidv 2794	. 2 ⊢ (𝐹 defAt 𝐴 → {𝑦 ∣ 𝑦 = (𝐹''''𝐴)} = {𝑦 ∣ 𝐴𝐹𝑦})
6		df-sn 4625	. . 3 ⊢ {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)}
7	6	a1i 11	. 2 ⊢ (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦 ∣ 𝑦 = (𝐹''''𝐴)})
8		dfdfat2 46571	. . 3 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9		imasng 6082	. . . 4 ⊢ (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
10	9	adantr 479	. . 3 ⊢ ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹 “ {𝐴}) = {𝑦 ∣ 𝐴𝐹𝑦})
11	8, 10	sylbi 216	. 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 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∃!weu 2556  {cab 2702  Vcvv 3463  {csn 4624   class class class wbr 5143  dom cdm 5672   “ cima 5675   defAt wdfat 46559  ''''cafv2 46651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-dfat 46562  df-afv2 46652

This theorem is referenced by:  afv2co2  46700