Theorem dfafv23 48596

Description: A definition of function value in terms of iota, analogous to dffv3 6912. (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 48553	. 2 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2		dfdfat2 48471	. . . . . . 7 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
3	2	simplbi 502	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → 𝐴 ∈ dom 𝐹)
4		elimasng 6105	. . . . . 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 5109	. . . . 5 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
7	5, 6	bitr4di 292	. . . 4 ⊢ ((𝐹 defAt 𝐴 ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
8	7	elvd 3460	. . 3 ⊢ (𝐹 defAt 𝐴 → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
9	8	iotabidv 6549	. 2 ⊢ (𝐹 defAt 𝐴 → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
10	1, 9	eqtr4d 2801	1 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃!weu 2596  Vcvv 3454  {csn 4588  ⟨cop 4840   class class class wbr 5108  dom cdm 5658   “ cima 5661  ℩cio 6518   defAt wdfat 48460  ''''cafv2 48551

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-dfat 48463  df-afv2 48552

This theorem is used by:  afv2co2  48600