Theorem opabiota 6911

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 6829	. . 3 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
2		opabiota.2	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	iotabidv 6471	. . 3 ⊢ (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
4	1, 3	eqeq12d 2751	. 2 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = (℩𝑦𝜑) ↔ (𝐹‘𝐵) = (℩𝑦𝜓)))
5		vex 3431	. . . 4 ⊢ 𝑥 ∈ V
6	5	eldm 5844	. . 3 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7		nfiota1 6445	. . . . 5 ⊢ Ⅎ𝑦(℩𝑦𝜑)
8	7	nfeq2 2914	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (℩𝑦𝜑)
9		opabiota.1	. . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
10	9	opabiotafun 6909	. . . . . 6 ⊢ Fun 𝐹
11		funbrfv 6877	. . . . . 6 ⊢ (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦))
12	10, 11	ax-mp 5	. . . . 5 ⊢ (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦)
13		df-br 5075	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
14	9	eleq2i 2827	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
15		opabidw 5468	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ {𝑦 ∣ 𝜑} = {𝑦})
16	13, 14, 15	3bitri 297	. . . . . . 7 ⊢ (𝑥𝐹𝑦 ↔ {𝑦 ∣ 𝜑} = {𝑦})
17		vsnid 4597	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
18		id 22	. . . . . . . . 9 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → {𝑦 ∣ 𝜑} = {𝑦})
19	17, 18	eleqtrrid 2842	. . . . . . . 8 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → 𝑦 ∈ {𝑦 ∣ 𝜑})
20		abid 2717	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
21	19, 20	sylib 218	. . . . . . 7 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → 𝜑)
22	16, 21	sylbi 217	. . . . . 6 ⊢ (𝑥𝐹𝑦 → 𝜑)
23		vex 3431	. . . . . . . . 9 ⊢ 𝑦 ∈ V
24	5, 23	breldm 5852	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 → 𝑥 ∈ dom 𝐹)
25	9	opabiotadm 6910	. . . . . . . . 9 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
26	25	eqabri 2877	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
27	24, 26	sylib 218	. . . . . . 7 ⊢ (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28		iota1 6466	. . . . . . 7 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
29	27, 28	syl 17	. . . . . 6 ⊢ (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
30	22, 29	mpbid 232	. . . . 5 ⊢ (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
31	12, 30	eqtr4d 2773	. . . 4 ⊢ (𝑥𝐹𝑦 → (𝐹‘𝑥) = (℩𝑦𝜑))
32	8, 31	exlimi 2224	. . 3 ⊢ (∃𝑦 𝑥𝐹𝑦 → (𝐹‘𝑥) = (℩𝑦𝜑))
33	6, 32	sylbi 217	. 2 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = (℩𝑦𝜑))
34	4, 33	vtoclga 3518	1 ⊢ (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = (℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2567  {cab 2713  {csn 4557  ⟨cop 4563   class class class wbr 5074  {copab 5136  dom cdm 5620  ℩cio 6441  Fun wfun 6481  ‘cfv 6487

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6443  df-fun 6489  df-fv 6495

This theorem is referenced by: (None)