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 3492	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-fv 6548	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑧𝐴𝐹𝑧)
3		breq2 5151	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦))
4		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑦𝐴
5		fvopab5.1	. . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6		nfopab2 5218	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
7	5, 6	nfcxfr 2901	. . . . . 6 ⊢ Ⅎ𝑦𝐹
8		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑦𝑧
9	4, 7, 8	nfbr 5194	. . . . 5 ⊢ Ⅎ𝑦 𝐴𝐹𝑧
10		nfv 1917	. . . . 5 ⊢ Ⅎ𝑧 𝐴𝐹𝑦
11	3, 9, 10	cbviotaw 6499	. . . 4 ⊢ (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
12	2, 11	eqtri 2760	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
13		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
14		nfopab1 5217	. . . . . . . 8 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
15	5, 14	nfcxfr 2901	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
16		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
17	13, 15, 16	nfbr 5194	. . . . . 6 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
18		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥𝜓
19	17, 18	nfbi 1906	. . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦 ↔ 𝜓)
20		breq1 5150	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
21		fvopab5.2	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
22	20, 21	bibi12d 345	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ↔ 𝜑) ↔ (𝐴𝐹𝑦 ↔ 𝜓)))
23		df-br 5148	. . . . . 6 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
24	5	eleq2i 2825	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25		opabidw 5523	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
26	23, 24, 25	3bitri 296	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 𝜑)
27	19, 22, 26	vtoclg1f 3555	. . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑦 ↔ 𝜓))
28	27	iotabidv 6524	. . 3 ⊢ (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
29	12, 28	eqtrid 2784	. 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑦𝜓))
30	1, 29	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1541   ∈ wcel 2106  Vcvv 3474  ⟨cop 4633   class class class wbr 5147  {copab 5209  ℩cio 6490  ‘cfv 6540

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-iota 6492  df-fv 6548

This theorem is referenced by:  ajval  30101  adjval  31130