Theorem fvopab5 6907

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 3450	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-fv 6441	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑧𝐴𝐹𝑧)
3		breq2 5078	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦))
4		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦𝐴
5		fvopab5.1	. . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6		nfopab2 5145	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
7	5, 6	nfcxfr 2905	. . . . . 6 ⊢ Ⅎ𝑦𝐹
8		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦𝑧
9	4, 7, 8	nfbr 5121	. . . . 5 ⊢ Ⅎ𝑦 𝐴𝐹𝑧
10		nfv 1917	. . . . 5 ⊢ Ⅎ𝑧 𝐴𝐹𝑦
11	3, 9, 10	cbviotaw 6398	. . . 4 ⊢ (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
12	2, 11	eqtri 2766	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
13		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
14		nfopab1 5144	. . . . . . . 8 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
15	5, 14	nfcxfr 2905	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
16		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
17	13, 15, 16	nfbr 5121	. . . . . 6 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
18		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥𝜓
19	17, 18	nfbi 1906	. . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦 ↔ 𝜓)
20		breq1 5077	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐹𝑦 ↔ 𝐴𝐹𝑦))
21		fvopab5.2	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
22	20, 21	bibi12d 346	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐹𝑦 ↔ 𝜑) ↔ (𝐴𝐹𝑦 ↔ 𝜓)))
23		df-br 5075	. . . . . 6 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
24	5	eleq2i 2830	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25		opabidw 5437	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
26	23, 24, 25	3bitri 297	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 𝜑)
27	19, 22, 26	vtoclg1f 3504	. . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑦 ↔ 𝜓))
28	27	iotabidv 6417	. . 3 ⊢ (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
29	12, 28	eqtrid 2790	. 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑦𝜓))
30	1, 29	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567   class class class wbr 5074  {copab 5136  ℩cio 6389  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-iota 6391  df-fv 6441

This theorem is referenced by:  ajval  29223  adjval  30252