Theorem dfafv23 47358

Description: A definition of function value in terms of iota, analogous to dffv3 6824. (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 47315	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		dfdfat2 47233	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
3	2	simplbi 497	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → 𝐴 ∈ dom 𝐹)
4		elimasng 6043	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
5	3, 4	sylan 580	. . . . 5 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
6		df-br 5094	. . . . 5 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
7	5, 6	bitr4di 289	. . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
8	7	elvd 3442	. . 3 ⊢ (𝐹 defAt 𝐴 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
9	8	iotabidv 6471	. 2 ⊢ (𝐹 defAt 𝐴 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
10	1, 9	eqtr4d 2769	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃!weu 2563  Vcvv 3436  {csn 4575  ⟨cop 4581   class class class wbr 5093  dom cdm 5619   “ cima 5622  ℩cio 6441   defAt wdfat 47221  ''''cafv2 47313

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372

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-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-dfat 47224  df-afv2 47314

This theorem is referenced by:  afv2co2  47362