Theorem dfafv23 45010

Description: A definition of function value in terms of iota, analogous to dffv3 6808. (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 44967	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		dfdfat2 44885	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
3	2	simplbi 498	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → 𝐴 ∈ dom 𝐹)
4		elimasng 6014	. . . . . 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 5088	. . . . 5 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
7	5, 6	bitr4di 288	. . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
8	7	elvd 3448	. . 3 ⊢ (𝐹 defAt 𝐴 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
9	8	iotabidv 6450	. 2 ⊢ (𝐹 defAt 𝐴 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
10	1, 9	eqtr4d 2780	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∃!weu 2567  Vcvv 3441  {csn 4571  ⟨cop 4577   class class class wbr 5087  dom cdm 5608   “ cima 5611  ℩cio 6416   defAt wdfat 44873  ''''cafv2 44965

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-id 5507  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-iota 6418  df-fun 6468  df-dfat 44876  df-afv2 44966

This theorem is referenced by:  afv2co2  45014