Theorem dffv3 6668

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		elimasng 5957	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
2		df-br 5069	. . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
3	1, 2	syl6bbr 291	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
4	3	elvd 3502	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
5	4	iotabidv 6341	. . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
6		df-fv 6365	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
7	5, 6	syl6reqr 2877	. 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
8		fvprc 6665	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
9		snprc 4655	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
10	9	biimpi 218	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
11	10	imaeq2d 5931	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
12		ima0 5947	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
13	11, 12	syl6eq 2874	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
14	13	eleq2d 2900	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
15	14	iotabidv 6341	. . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
16		noel 4298	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
17	16	nex 1801	. . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅
18		euex 2662	. . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
19	17, 18	mto 199	. . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅
20		iotanul 6335	. . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
21	19, 20	ax-mp 5	. . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅
22	15, 21	syl6eq 2874	. . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
23	8, 22	eqtr4d 2861	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
24	7, 23	pm2.61i 184	1 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))

Syntax hints:  ¬ wn 3   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃!weu 2653  Vcvv 3496  ∅c0 4293  {csn 4569  ⟨cop 4575   class class class wbr 5068   “ cima 5560  ℩cio 6314  ‘cfv 6357

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fv 6365

This theorem is referenced by:  dffv4  6669  fvco2  6760  shftval  14435  dffv5  33387