Theorem opabiota 6987

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 6905	. . 3 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
2		opabiota.2	. . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓))
3	2	iotabidv 6538	. . 3 ⊢ (𝑥 = 𝐵 → (℩𝑦𝜑) = (℩𝑦𝜓))
4	1, 3	eqeq12d 2779	. 2 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = (℩𝑦𝜑) ↔ (𝐹‘𝐵) = (℩𝑦𝜓)))
5		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
6	5	eldm 5888	. . 3 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
7		nfiota1 6511	. . . . 5 ⊢ Ⅎ𝑦(℩𝑦𝜑)
8	7	nfeq2 2943	. . . 4 ⊢ Ⅎ𝑦(𝐹‘𝑥) = (℩𝑦𝜑)
9		opabiota.1	. . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
10	9	opabiotafun 6985	. . . . . 6 ⊢ Fun 𝐹
11		funbrfv 6953	. . . . . 6 ⊢ (Fun 𝐹 → (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦))
12	10, 11	ax-mp 5	. . . . 5 ⊢ (𝑥𝐹𝑦 → (𝐹‘𝑥) = 𝑦)
13		df-br 5107	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
14	9	eleq2i 2856	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}})
15		opabidw 5502	. . . . . . . 8 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} ↔ {𝑦 ∣ 𝜑} = {𝑦})
16	13, 14, 15	3bitri 300	. . . . . . 7 ⊢ (𝑥𝐹𝑦 ↔ {𝑦 ∣ 𝜑} = {𝑦})
17		vsnid 4624	. . . . . . . . 9 ⊢ 𝑦 ∈ {𝑦}
18		id 23	. . . . . . . . 9 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → {𝑦 ∣ 𝜑} = {𝑦})
19	17, 18	eleqtrrid 2871	. . . . . . . 8 ⊢ ({𝑦 ∣ 𝜑} = {𝑦} → 𝑦 ∈ {𝑦 ∣ 𝜑})
20		abid 2745	. . . . . . . 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 3458	. . . . . . . . 9 ⊢ 𝑦 ∈ V
24	5, 23	breldm 5896	. . . . . . . 8 ⊢ (𝑥𝐹𝑦 → 𝑥 ∈ dom 𝐹)
25	9	opabiotadm 6986	. . . . . . . . 9 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
26	25	eqabri 2906	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃!𝑦𝜑)
27	24, 26	sylib 221	. . . . . . 7 ⊢ (𝑥𝐹𝑦 → ∃!𝑦𝜑)
28		iota1 6533	. . . . . . 7 ⊢ (∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
29	27, 28	syl 18	. . . . . 6 ⊢ (𝑥𝐹𝑦 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦))
30	22, 29	mpbid 235	. . . . 5 ⊢ (𝑥𝐹𝑦 → (℩𝑦𝜑) = 𝑦)
31	12, 30	eqtr4d 2801	. . . 4 ⊢ (𝑥𝐹𝑦 → (𝐹‘𝑥) = (℩𝑦𝜑))
32	8, 31	exlimi 2257	. . 3 ⊢ (∃𝑦 𝑥𝐹𝑦 → (𝐹‘𝑥) = (℩𝑦𝜑))
33	6, 32	sylbi 220	. 2 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = (℩𝑦𝜑))
34	4, 33	vtoclga 3540	1 ⊢ (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = (℩𝑦𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596  {cab 2741  {csn 4588  ⟨cop 4838   class class class wbr 5106  {copab 5170  dom cdm 5655  ℩cio 6507  Fun wfun 6549  ‘cfv 6555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6509  df-fun 6557  df-fv 6563

This theorem is used by: (None)