Theorem dffv3 6878

Description: A definition of function value in terms of iota. (Contributed by Scott Fenton, 19-Feb-2013.)

Assertion

Ref	Expression
dffv3	⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐴

Proof of Theorem dffv3

Step	Hyp	Ref	Expression
1		df-fv 6545	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		elimasng 6089	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3		df-br 5108	. . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
4	2, 3	bitr4di 292	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
5	4	elvd 3459	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
6	5	iotabidv 6521	. . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
7	1, 6	eqtr4id 2816	. 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
8		fvprc 6874	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
9		snprc 4681	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 219	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	imaeq2d 6060	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
12		ima0 6077	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
13	11, 12	eqtrdi 2813	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
14	13	eleq2d 2848	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
15	14	iotabidv 6521	. . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
16		noel 4287	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
17	16	nex 1833	. . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅
18		euex 2604	. . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
19	17, 18	mto 200	. . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅
20		iotanul 6517	. . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
21	19, 20	ax-mp 5	. . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅
22	15, 21	eqtrdi 2813	. . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
23	8, 22	eqtr4d 2800	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
24	7, 23	pm2.61i 184	1 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2595  Vcvv 3453  ∅c0 4282  {csn 4587  ⟨cop 4593   class class class wbr 5107   “ cima 5662  ℩cio 6491  ‘cfv 6537

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

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-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fv 6545

This theorem is used by:  dffv4  6879  fvco2  6979  shftval  15151  dffv5  36509