Theorem opabex3 7950

Description: Existence of an ordered pair abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
opabex3.1	⊢ 𝐴 ∈ V
opabex3.2	⊢ (𝑥 ∈ 𝐴 → {𝑦 ∣ 𝜑} ∈ V)

Assertion

Ref	Expression
opabex3	⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabex3

Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.42v 1949	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2		an12 642	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	exbii 1842	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
4		elxp 5692	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
5		excom 2154	. . . . . . . . 9 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
6		an12 642	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
7		velsn 4639	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
8	7	anbi1i 623	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
9	6, 8	bitri 275	. . . . . . . . . . . 12 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
10	9	exbii 1842	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
11		opeq1 4868	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
12	11	eqeq2d 2737	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
13	12	anbi1d 629	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
14	13	equsexvw 2000	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
15	10, 14	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
16	15	exbii 1842	. . . . . . . . 9 ⊢ (∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
17	5, 16	bitri 275	. . . . . . . 8 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
18		nfv 1909	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑧 = ⟨𝑥, 𝑤⟩
19		nfsab1 2711	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ {𝑦 ∣ 𝜑}
20	18, 19	nfan 1894	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})
21		nfv 1909	. . . . . . . . 9 ⊢ Ⅎ𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
22		opeq2 4869	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
23	22	eqeq2d 2737	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
24		df-clab 2704	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑦 ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
25		sbequ12 2235	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
26	25	equcoms 2015	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
27	24, 26	bitr4id 290	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑))
28	23, 27	anbi12d 630	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
29	20, 21, 28	cbvexv1 2332	. . . . . . . 8 ⊢ (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
30	4, 17, 29	3bitri 297	. . . . . . 7 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
31	30	anbi2i 622	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
32	1, 3, 31	3bitr4ri 304	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
33	32	exbii 1842	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
34		eliun 4994	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}))
35		df-rex 3065	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})))
36	34, 35	bitri 275	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})))
37		elopab 5520	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
38	33, 36, 37	3bitr4i 303	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
39	38	eqriv 2723	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
40		opabex3.1	. . 3 ⊢ 𝐴 ∈ V
41		vsnex 5422	. . . . 5 ⊢ {𝑥} ∈ V
42		opabex3.2	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∣ 𝜑} ∈ V)
43		xpexg 7733	. . . . 5 ⊢ (({𝑥} ∈ V ∧ {𝑦 ∣ 𝜑} ∈ V) → ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
44	41, 42, 43	sylancr 586	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
45	44	rgen 3057	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V
46		iunexg 7946	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
47	40, 45, 46	mp2an 689	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V
48	39, 47	eqeltrri 2824	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wex 1773  [wsb 2059   ∈ wcel 2098  {cab 2703  ∀wral 3055  ∃wrex 3064  Vcvv 3468  {csn 4623  ⟨cop 4629  ∪ ciun 4990  {copab 5203   × cxp 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-opab 5204  df-xp 5675  df-rel 5676

This theorem is referenced by:  dvdsrval  20260  eulerpartlemgvv  33904