Theorem dffv3 6146

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		vex 3194	. . . . 5 ⊢ 𝑥 ∈ V
2		elimasng 5454	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹))
3		df-br 4619	. . . . . 6 ⊢ (𝐴𝐹𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐹)
4	2, 3	syl6bbr 278	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝑥 ∈ V) → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
5	1, 4	mpan2 706	. . . 4 ⊢ (𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝐴𝐹𝑥))
6	5	iotabidv 5834	. . 3 ⊢ (𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝐴𝐹𝑥))
7		df-fv 5858	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑥𝐴𝐹𝑥)
8	6, 7	syl6reqr 2679	. 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
9		fvprc 6144	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = ∅)
10		snprc 4228	. . . . . . . . 9 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 206	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	imaeq2d 5429	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = (𝐹 “ ∅))
13		ima0 5444	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
14	12, 13	syl6eq 2676	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝐹 “ {𝐴}) = ∅)
15	14	eleq2d 2689	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (𝑥 ∈ (𝐹 “ {𝐴}) ↔ 𝑥 ∈ ∅))
16	15	iotabidv 5834	. . . 4 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = (℩𝑥𝑥 ∈ ∅))
17		noel 3900	. . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅
18	17	nex 1728	. . . . . 6 ⊢ ¬ ∃𝑥 𝑥 ∈ ∅
19		euex 2498	. . . . . 6 ⊢ (∃!𝑥 𝑥 ∈ ∅ → ∃𝑥 𝑥 ∈ ∅)
20	18, 19	mto 188	. . . . 5 ⊢ ¬ ∃!𝑥 𝑥 ∈ ∅
21		iotanul 5828	. . . . 5 ⊢ (¬ ∃!𝑥 𝑥 ∈ ∅ → (℩𝑥𝑥 ∈ ∅) = ∅)
22	20, 21	ax-mp 5	. . . 4 ⊢ (℩𝑥𝑥 ∈ ∅) = ∅
23	16, 22	syl6eq 2676	. . 3 ⊢ (¬ 𝐴 ∈ V → (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})) = ∅)
24	9, 23	eqtr4d 2663	. 2 ⊢ (¬ 𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴})))
25	8, 24	pm2.61i 176	1 ⊢ (𝐹‘𝐴) = (℩𝑥𝑥 ∈ (𝐹 “ {𝐴}))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1992  ∃!weu 2474  Vcvv 3191  ∅c0 3896  {csn 4153  ⟨cop 4159   class class class wbr 4618   “ cima 5082  ℩cio 5811  ‘cfv 5850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fv 5858

This theorem is referenced by:  dffv4  6147  fvco2  6231  shftval  13743  dffv5  31665