Theorem fvopab5 7004

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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		df-fv 6522	. . . 4 ⊢ (𝐹‘𝐴) = (℩𝑧𝐴𝐹𝑧)
3		breq2 5114	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧 ↔ 𝐴𝐹𝑦))
4		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑦𝐴
5		fvopab5.1	. . . . . . 7 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6		nfopab2 5181	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
7	5, 6	nfcxfr 2890	. . . . . 6 ⊢ Ⅎ𝑦𝐹
8		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑦𝑧
9	4, 7, 8	nfbr 5157	. . . . 5 ⊢ Ⅎ𝑦 𝐴𝐹𝑧
10		nfv 1914	. . . . 5 ⊢ Ⅎ𝑧 𝐴𝐹𝑦
11	3, 9, 10	cbviotaw 6474	. . . 4 ⊢ (℩𝑧𝐴𝐹𝑧) = (℩𝑦𝐴𝐹𝑦)
12	2, 11	eqtri 2753	. . 3 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
13		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
14		nfopab1 5180	. . . . . . . 8 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
15	5, 14	nfcxfr 2890	. . . . . . 7 ⊢ Ⅎ𝑥𝐹
16		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
17	13, 15, 16	nfbr 5157	. . . . . 6 ⊢ Ⅎ𝑥 𝐴𝐹𝑦
18		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥𝜓
19	17, 18	nfbi 1903	. . . . 5 ⊢ Ⅎ𝑥(𝐴𝐹𝑦 ↔ 𝜓)
20		breq1 5113	. . . . . 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 5111	. . . . . 6 ⊢ (𝑥𝐹𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐹)
24	5	eleq2i 2821	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
25		opabidw 5487	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
26	23, 24, 25	3bitri 297	. . . . 5 ⊢ (𝑥𝐹𝑦 ↔ 𝜑)
27	19, 22, 26	vtoclg1f 3539	. . . 4 ⊢ (𝐴 ∈ V → (𝐴𝐹𝑦 ↔ 𝜓))
28	27	iotabidv 6498	. . 3 ⊢ (𝐴 ∈ V → (℩𝑦𝐴𝐹𝑦) = (℩𝑦𝜓))
29	12, 28	eqtrid 2777	. 2 ⊢ (𝐴 ∈ V → (𝐹‘𝐴) = (℩𝑦𝜓))
30	1, 29	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → (𝐹‘𝐴) = (℩𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598   class class class wbr 5110  {copab 5172  ℩cio 6465  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-iota 6467  df-fv 6522

This theorem is referenced by:  ajval  30797  adjval  31826