Theorem opabex3d 7919

Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 9-Aug-2024.)

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)   𝑉(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 1955	. . . . . 6 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2		an12 646	. . . . . . 7 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3	2	exbii 1850	. . . . . 6 ⊢ (∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4		elxp 5655	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
5		excom 2168	. . . . . . . . 9 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
6		an12 646	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
7		velsn 4598	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑥} ↔ 𝑣 = 𝑥)
8	7	anbi1i 625	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ {𝑥} ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
9	6, 8	bitri 275	. . . . . . . . . . . 12 ⊢ ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
10	9	exbii 1850	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ ∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
11		opeq1 4831	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑥 → ⟨𝑣, 𝑤⟩ = ⟨𝑥, 𝑤⟩)
12	11	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑥 → (𝑧 = ⟨𝑣, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑤⟩))
13	12	anbi1d 632	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑥 → ((𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})))
14	13	equsexvw 2007	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑣 = 𝑥 ∧ (𝑧 = ⟨𝑣, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}))
15	10, 14	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ (𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}))
16	15	exbii 1850	. . . . . . . . 9 ⊢ (∃𝑤∃𝑣(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}))
17	5, 16	bitri 275	. . . . . . . 8 ⊢ (∃𝑣∃𝑤(𝑧 = ⟨𝑣, 𝑤⟩ ∧ (𝑣 ∈ {𝑥} ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}))
18		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑧 = ⟨𝑥, 𝑤⟩
19		nfsab1 2723	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑤 ∈ {𝑦 ∣ 𝜓}
20	18, 19	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓})
21		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
22		opeq2 4832	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → ⟨𝑥, 𝑤⟩ = ⟨𝑥, 𝑦⟩)
23	22	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑧 = ⟨𝑥, 𝑤⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
24		df-clab 2716	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑦 ∣ 𝜓} ↔ [𝑤 / 𝑦]𝜓)
25		sbequ12 2259	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
26	25	equcoms 2022	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑦 → (𝜓 ↔ [𝑤 / 𝑦]𝜓))
27	24, 26	bitr4id 290	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ {𝑦 ∣ 𝜓} ↔ 𝜓))
28	23, 27	anbi12d 633	. . . . . . . . 9 ⊢ (𝑤 = 𝑦 → ((𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
29	20, 21, 28	cbvexv1 2347	. . . . . . . 8 ⊢ (∃𝑤(𝑧 = ⟨𝑥, 𝑤⟩ ∧ 𝑤 ∈ {𝑦 ∣ 𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
30	4, 17, 29	3bitri 297	. . . . . . 7 ⊢ (𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
31	30	anbi2i 624	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
32	1, 3, 31	3bitr4ri 304	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})) ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
33	32	exbii 1850	. . . 4 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
34		eliun 4952	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓}))
35		df-rex 3063	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})))
36	34, 35	bitri 275	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥} × {𝑦 ∣ 𝜓})))
37		elopab 5483	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)))
38	33, 36, 37	3bitr4i 303	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)})
39	38	eqriv 2734	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)}
40		opabex3d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
41		vsnex 5381	. . . . 5 ⊢ {𝑥} ∈ V
42		opabex3d.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝜓} ∈ V)
43		xpexg 7705	. . . . 5 ⊢ (({𝑥} ∈ V ∧ {𝑦 ∣ 𝜓} ∈ V) → ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
44	41, 42, 43	sylancr 588	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
45	44	ralrimiva 3130	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
46		iunexg 7917	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V) → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
47	40, 45, 46	syl2anc 585	. 2 ⊢ (𝜑 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝑦 ∣ 𝜓}) ∈ V)
48	39, 47	eqeltrrid 2842	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781  [wsb 2068   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3442  {csn 4582  ⟨cop 4588  ∪ ciun 4948  {copab 5162   × cxp 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-opab 5163  df-xp 5638  df-rel 5639

This theorem is referenced by:  wksfval  29695  fpwrelmap  32822  satfv0  35571  cnvepresex  38581  tfsconcatun  43688  opabresex0d  47639  upwlksfval  48489  upfval2  49530