Theorem opabiota 6574

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 6499	. . 3 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
2		opabiota.2	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	iotabidv 6173	. . 3 ⊢ (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
4	1, 3	eqeq12d 2794	. 2 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = (℩𝑦𝜑) ↔ (𝐹‘𝐵) = (℩𝑦𝜓)))
5		vex 3419	. . . 4 ⊢ 𝑥 ∈ V
6	5	eldm 5619	. . 3 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7		nfiota1 6154	. . . . 5 ⊢ Ⅎ𝑦(℩𝑦𝜑)
8	7	nfeq2 2948	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (℩𝑦𝜑)
9		opabiota.1	. . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
10	9	opabiotafun 6572	. . . . . 6 ⊢ Fun 𝐹
11		funbrfv 6546	. . . . . 6 ⊢ (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦))
12	10, 11	ax-mp 5	. . . . 5 ⊢ (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦)
13		df-br 4930	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
14	9	eleq2i 2858	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
15		opabid 5268	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ {𝑦 ∣ 𝜑} = {𝑦})
16	13, 14, 15	3bitri 289	. . . . . . 7 ⊢ (𝑥𝐹𝑦 ↔ {𝑦 ∣ 𝜑} = {𝑦})
17		vsnid 4474	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
18		id 22	. . . . . . . . 9 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → {𝑦 ∣ 𝜑} = {𝑦})
19	17, 18	syl5eleqr 2874	. . . . . . . 8 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → 𝑦 ∈ {𝑦 ∣ 𝜑})
20		abid 2763	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
21	19, 20	sylib 210	. . . . . . 7 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → 𝜑)
22	16, 21	sylbi 209	. . . . . 6 ⊢ (𝑥𝐹𝑦 → 𝜑)
23		vex 3419	. . . . . . . . 9 ⊢ 𝑦 ∈ V
24	5, 23	breldm 5627	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 → 𝑥 ∈ dom 𝐹)
25	9	opabiotadm 6573	. . . . . . . . 9 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
26	25	abeq2i 2901	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
27	24, 26	sylib 210	. . . . . . 7 ⊢ (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28		iota1 6166	. . . . . . 7 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
29	27, 28	syl 17	. . . . . 6 ⊢ (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
30	22, 29	mpbid 224	. . . . 5 ⊢ (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
31	12, 30	eqtr4d 2818	. . . 4 ⊢ (𝑥𝐹𝑦 → (𝐹‘𝑥) = (℩𝑦𝜑))
32	8, 31	exlimi 2147	. . 3 ⊢ (∃𝑦 𝑥𝐹𝑦 → (𝐹‘𝑥) = (℩𝑦𝜑))
33	6, 32	sylbi 209	. 2 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = (℩𝑦𝜑))
34	4, 33	vtoclga 3494	1 ⊢ (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = (℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1507  ∃wex 1742   ∈ wcel 2050  ∃!weu 2583  {cab 2759  {csn 4441  ⟨cop 4447   class class class wbr 4929  {copab 4991  dom cdm 5407  ℩cio 6150  Fun wfun 6182  ‘cfv 6188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pr 5186

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196

This theorem is referenced by: (None)