Theorem opabex3d 6027

Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)

Hypotheses

Ref	Expression
opabex3d.1	⊢ (𝜑 → 𝐴 ∈ V)
opabex3d.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝜓} ∈ V)

Assertion

Ref	Expression
opabex3d	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ∈ V)

Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem opabex3d

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

Step	Hyp	Ref	Expression
1		19.42v 1879	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2		an12 551	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3	2	exbii 1585	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4		elxp 4564	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
5		excom 1643	. . . . . . . . 9 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
6		an12 551	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
7		velsn 3549	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
8	7	anbi1i 454	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
9	6, 8	bitri 183	. . . . . . . . . . . 12 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
10	9	exbii 1585	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
11		vex 2692	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		opeq1 3713	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
13	12	eqeq2d 2152	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
14	13	anbi1d 461	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
15	11, 14	ceqsexv 2728	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}))
16	10, 15	bitri 183	. . . . . . . . . 10 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}))
17	16	exbii 1585	. . . . . . . . 9 ⊢ (∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}))
18	5, 17	bitri 183	. . . . . . . 8 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}))
19		nfv 1509	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑧 = ⟨𝑥, 𝑤⟩
20		nfsab1 2130	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ {𝑦 ∣ 𝜓}
21	19, 20	nfan 1545	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})
22		nfv 1509	. . . . . . . . 9 ⊢ Ⅎ𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
23		opeq2 3714	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
24	23	eqeq2d 2152	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
25		df-clab 2127	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑦 ∣ 𝜓} ↔ [𝑤 / 𝑦]𝜓)
26		sbequ12 1745	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
27	26	equcoms 1685	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
28	25, 27	bitr4id 198	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝑦 ∣ 𝜓} ↔ 𝜓))
29	24, 28	anbi12d 465	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
30	21, 22, 29	cbvex 1730	. . . . . . . 8 ⊢ (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
31	4, 18, 30	3bitri 205	. . . . . . 7 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
32	31	anbi2i 453	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
33	1, 3, 32	3bitr4ri 212	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
34	33	exbii 1585	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
35		eliun 3825	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓}))
36		df-rex 2423	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})))
37	35, 36	bitri 183	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})))
38		elopab 4188	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
39	34, 37, 38	3bitr4i 211	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
40	39	eqriv 2137	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
41		opabex3d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ V)
42		snexg 4116	. . . . . 6 ⊢ (𝑥 ∈ V → {𝑥} ∈ V)
43	11, 42	ax-mp 5	. . . . 5 ⊢ {𝑥} ∈ V
44		opabex3d.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝜓} ∈ V)
45		xpexg 4661	. . . . 5 ⊢ (({𝑥} ∈ V ∧ {𝑦 ∣ 𝜓} ∈ V) → ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
46	43, 44, 45	sylancr 411	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
47	46	ralrimiva 2508	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
48		iunexg 6025	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
49	41, 47, 48	syl2anc 409	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
50	40, 49	eqeltrrid 2228	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332  ∃wex 1469   ∈ wcel 1481  [wsb 1736  {cab 2126  ∀wral 2417  ∃wrex 2418  Vcvv 2689  {csn 3532  ⟨cop 3535  ∪ ciun 3821  {copab 3996   × cxp 4545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139

This theorem is referenced by:  acfun  7080  ccfunen  7096  ovshftex  10623