Theorem dfafv23 48459

Description: A definition of function value in terms of iota, analogous to dffv3 6886. (Contributed by AV, 6-Sep-2022.)

Assertion

Ref	Expression
dfafv23	⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem dfafv23

Step	Hyp	Ref	Expression
1		dfatafv2iota 48416	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		dfdfat2 48334	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
3	2	simplbi 502	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → 𝐴 ∈ dom 𝐹)
4		elimasng 6089	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
5	3, 4	sylan 592	. . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
6		df-br 5103	. . . . 5 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
7	5, 6	bitr4di 292	. . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
8	7	elvd 3456	. . 3 ⊢ (𝐹 defAt 𝐴 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
9	8	iotabidv 6526	. 2 ⊢ (𝐹 defAt 𝐴 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
10	1, 9	eqtr4d 2798	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∃!weu 2593  Vcvv 3450  {csn 4584  ⟨cop 4834   class class class wbr 5102  dom cdm 5651   “ cima 5654  ℩cio 6495   defAt wdfat 48323  ''''cafv2 48414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6497  df-fun 6545  df-dfat 48326  df-afv2 48415

This theorem is used by:  afv2co2  48463