Theorem fvopab5 7027

Description: The value of a function that is expressed as an ordered pair abstraction. (Contributed by NM, 19-Feb-2006.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypotheses

Ref	Expression
fvopab5.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
fvopab5.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
fvopab5	⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝜓,𝑥

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

Proof of Theorem fvopab5

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3478	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-fv 6548	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑧𝐴𝐹𝑧)
3		breq2 5115	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦))
4		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑦𝐴
5		fvopab5.1	. . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6		nfopab2 5184	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
7	5, 6	nfcxfr 2925	. . . . . 6 ⊢ Ⅎ𝑦𝐹
8		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑦𝑧
9	4, 7, 8	nfbr 5160	. . . . 5 ⊢ Ⅎ𝑦 𝐴𝐹𝑧
10		nfv 1947	. . . . 5 ⊢ Ⅎ𝑧 𝐴𝐹𝑦
11	3, 9, 10	cbviotaw 6503	. . . 4 ⊢ (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
12	2, 11	eqtri 2788	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
13		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
14		nfopab1 5183	. . . . . . . 8 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
15	5, 14	nfcxfr 2925	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
16		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
17	13, 15, 16	nfbr 5160	. . . . . 6 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
18		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥𝜓
19	17, 18	nfbi 1936	. . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦 ↔ 𝜓)
20		breq1 5114	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
21		fvopab5.2	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
22	20, 21	bibi12d 348	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ↔ 𝜑) ↔ (𝐴𝐹𝑦 ↔ 𝜓)))
23		df-br 5112	. . . . . 6 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
24	5	eleq2i 2857	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25		opabidw 5510	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
26	23, 24, 25	3bitri 300	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 𝜑)
27	19, 22, 26	vtoclg1f 3537	. . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑦 ↔ 𝜓))
28	27	iotabidv 6524	. . 3 ⊢ (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
29	12, 28	eqtrid 2812	. 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑦𝜓))
30	1, 29	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146  Vcvv 3457  ⟨cop 4597   class class class wbr 5111  {copab 5175  ℩cio 6494  ‘cfv 6540

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-iota 6496  df-fv 6548

This theorem is used by:  ajval  31286  adjval  32315