Theorem dffv3 6770

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 6441	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
2		elimasng 5996	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3		df-br 5075	. . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
4	2, 3	bitr4di 289	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
5	4	elvd 3439	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
6	5	iotabidv 6417	. . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
7	1, 6	eqtr4id 2797	. 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
8		fvprc 6766	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
9		snprc 4653	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 215	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	imaeq2d 5969	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
12		ima0 5985	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
13	11, 12	eqtrdi 2794	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
14	13	eleq2d 2824	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
15	14	iotabidv 6417	. . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
16		noel 4264	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
17	16	nex 1803	. . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅
18		euex 2577	. . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
19	17, 18	mto 196	. . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅
20		iotanul 6411	. . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
21	19, 20	ax-mp 5	. . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅
22	15, 21	eqtrdi 2794	. . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
23	8, 22	eqtr4d 2781	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
24	7, 23	pm2.61i 182	1 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃!weu 2568  Vcvv 3432  ∅c0 4256  {csn 4561  ⟨cop 4567   class class class wbr 5074   “ cima 5592  ℩cio 6389  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fv 6441

This theorem is referenced by:  dffv4  6771  fvco2  6865  shftval  14785  dffv5  34226