Theorem opabiota 6721

Description: Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)

Hypotheses

Ref	Expression
opabiota.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
opabiota.2	⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
opabiota	⊢ (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = (℩𝑦𝜓))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)

Proof of Theorem opabiota

Step	Hyp	Ref	Expression
1		fveq2 6645	. . 3 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
2		opabiota.2	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	iotabidv 6308	. . 3 ⊢ (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
4	1, 3	eqeq12d 2814	. 2 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = (℩𝑦𝜑) ↔ (𝐹‘𝐵) = (℩𝑦𝜓)))
5		vex 3444	. . . 4 ⊢ 𝑥 ∈ V
6	5	eldm 5733	. . 3 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7		nfiota1 6285	. . . . 5 ⊢ Ⅎ𝑦(℩𝑦𝜑)
8	7	nfeq2 2972	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (℩𝑦𝜑)
9		opabiota.1	. . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
10	9	opabiotafun 6719	. . . . . 6 ⊢ Fun 𝐹
11		funbrfv 6691	. . . . . 6 ⊢ (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦))
12	10, 11	ax-mp 5	. . . . 5 ⊢ (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦)
13		df-br 5031	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
14	9	eleq2i 2881	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
15		opabidw 5377	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ {𝑦 ∣ 𝜑} = {𝑦})
16	13, 14, 15	3bitri 300	. . . . . . 7 ⊢ (𝑥𝐹𝑦 ↔ {𝑦 ∣ 𝜑} = {𝑦})
17		vsnid 4562	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
18		id 22	. . . . . . . . 9 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → {𝑦 ∣ 𝜑} = {𝑦})
19	17, 18	eleqtrrid 2897	. . . . . . . 8 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → 𝑦 ∈ {𝑦 ∣ 𝜑})
20		abid 2780	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
21	19, 20	sylib 221	. . . . . . 7 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → 𝜑)
22	16, 21	sylbi 220	. . . . . 6 ⊢ (𝑥𝐹𝑦 → 𝜑)
23		vex 3444	. . . . . . . . 9 ⊢ 𝑦 ∈ V
24	5, 23	breldm 5741	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 → 𝑥 ∈ dom 𝐹)
25	9	opabiotadm 6720	. . . . . . . . 9 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
26	25	abeq2i 2925	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
27	24, 26	sylib 221	. . . . . . 7 ⊢ (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28		iota1 6301	. . . . . . 7 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
29	27, 28	syl 17	. . . . . 6 ⊢ (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
30	22, 29	mpbid 235	. . . . 5 ⊢ (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
31	12, 30	eqtr4d 2836	. . . 4 ⊢ (𝑥𝐹𝑦 → (𝐹‘𝑥) = (℩𝑦𝜑))
32	8, 31	exlimi 2215	. . 3 ⊢ (∃𝑦 𝑥𝐹𝑦 → (𝐹‘𝑥) = (℩𝑦𝜑))
33	6, 32	sylbi 220	. 2 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = (℩𝑦𝜑))
34	4, 33	vtoclga 3522	1 ⊢ (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = (℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃!weu 2628  {cab 2776  {csn 4525  ⟨cop 4531   class class class wbr 5030  {copab 5092  dom cdm 5519  ℩cio 6281  Fun wfun 6318  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332

This theorem is referenced by: (None)