Theorem opabex3 6117

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 1906	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
2		an12 561	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
3	2	exbii 1605	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
4		elxp 4640	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
5		excom 1664	. . . . . . . . 9 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
6		an12 561	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
7		velsn 3608	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
8	7	anbi1i 458	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
9	6, 8	bitri 184	. . . . . . . . . . . 12 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
10	9	exbii 1605	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
11		vex 2740	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		opeq1 3776	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
13	12	eqeq2d 2189	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
14	13	anbi1d 465	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})))
15	11, 14	ceqsexv 2776	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
16	10, 15	bitri 184	. . . . . . . . . 10 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
17	16	exbii 1605	. . . . . . . . 9 ⊢ (∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
18	5, 17	bitri 184	. . . . . . . 8 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}))
19		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑧 = ⟨𝑥, 𝑤⟩
20		nfsab1 2167	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ {𝑦 ∣ 𝜑}
21	19, 20	nfan 1565	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑})
22		nfv 1528	. . . . . . . . 9 ⊢ Ⅎ𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
23		opeq2 3777	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
24	23	eqeq2d 2189	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
25		df-clab 2164	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑦 ∣ 𝜑} ↔ [𝑤 / 𝑦]𝜑)
26		sbequ12 1771	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
27	26	equcoms 1708	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝜑 ↔ [𝑤 / 𝑦]𝜑))
28	25, 27	bitr4id 199	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑))
29	24, 28	anbi12d 473	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
30	21, 22, 29	cbvex 1756	. . . . . . . 8 ⊢ (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
31	4, 18, 30	3bitri 206	. . . . . . 7 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
32	31	anbi2i 457	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
33	1, 3, 32	3bitr4ri 213	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
34	33	exbii 1605	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
35		eliun 3888	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}))
36		df-rex 2461	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})))
37	35, 36	bitri 184	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜑})))
38		elopab 4255	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜑)))
39	34, 37, 38	3bitr4i 212	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)})
40	39	eqriv 2174	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
41		opabex3.1	. . 3 ⊢ 𝐴 ∈ V
42		snexg 4181	. . . . . 6 ⊢ (𝑥 ∈ V → {𝑥} ∈ V)
43	11, 42	ax-mp 5	. . . . 5 ⊢ {𝑥} ∈ V
44		opabex3.2	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → {𝑦 ∣ 𝜑} ∈ V)
45		xpexg 4737	. . . . 5 ⊢ (({𝑥} ∈ V ∧ {𝑦 ∣ 𝜑} ∈ V) → ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
46	43, 44, 45	sylancr 414	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
47	46	rgen 2530	. . 3 ⊢ ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V
48		iunexg 6114	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V)
49	41, 47, 48	mp2an 426	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜑}) ∈ V
50	40, 49	eqeltrri 2251	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492  [wsb 1762   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  Vcvv 2737  {csn 3591  ⟨cop 3594  ∪ ciun 3884  {copab 4060   × cxp 4621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220

This theorem is referenced by: (None)