Theorem opabex3 7740

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 1962	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2		an12 645	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	exbii 1855	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
4		elxp 5574	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
5		excom 2166	. . . . . . . . 9 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
6		an12 645	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
7		velsn 4557	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
8	7	anbi1i 627	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
9	6, 8	bitri 278	. . . . . . . . . . . 12 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
10	9	exbii 1855	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
11		opeq1 4784	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
12	11	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
13	12	anbi1d 633	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
14	13	equsexvw 2013	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
15	10, 14	bitri 278	. . . . . . . . . 10 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
16	15	exbii 1855	. . . . . . . . 9 ⊢ (∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
17	5, 16	bitri 278	. . . . . . . 8 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
18		nfv 1922	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑧 = ⟨𝑥, 𝑤⟩
19		nfsab1 2722	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ {𝑦 ∣ 𝜑}
20	18, 19	nfan 1907	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})
21		nfv 1922	. . . . . . . . 9 ⊢ Ⅎ𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
22		opeq2 4785	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
23	22	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
24		df-clab 2715	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑦 ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
25		sbequ12 2249	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
26	25	equcoms 2028	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
27	24, 26	bitr4id 293	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑))
28	23, 27	anbi12d 634	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
29	20, 21, 28	cbvexv1 2342	. . . . . . . 8 ⊢ (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
30	4, 17, 29	3bitri 300	. . . . . . 7 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
31	30	anbi2i 626	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
32	1, 3, 31	3bitr4ri 307	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
33	32	exbii 1855	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
34		eliun 4908	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}))
35		df-rex 3067	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})))
36	34, 35	bitri 278	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})))
37		elopab 5408	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
38	33, 36, 37	3bitr4i 306	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
39	38	eqriv 2734	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
40		opabex3.1	. . 3 ⊢ 𝐴 ∈ V
41		snex 5324	. . . . 5 ⊢ {𝑥} ∈ V
42		opabex3.2	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∣ 𝜑} ∈ V)
43		xpexg 7535	. . . . 5 ⊢ (({𝑥} ∈ V ∧ {𝑦 ∣ 𝜑} ∈ V) → ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
44	41, 42, 43	sylancr 590	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
45	44	rgen 3071	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V
46		iunexg 7736	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
47	40, 45, 46	mp2an 692	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V
48	39, 47	eqeltrri 2835	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787  [wsb 2070   ∈ wcel 2110  {cab 2714  ∀wral 3061  ∃wrex 3062  Vcvv 3408  {csn 4541  ⟨cop 4547  ∪ ciun 4904  {copab 5115   × cxp 5549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388

This theorem is referenced by:  dvdsrval  19663  eulerpartlemgvv  32055