Theorem opabex3rd 7963

Description: Existence of an ordered pair abstraction if the second components are elements of a set. (Contributed by AV, 17-Sep-2023.) (Revised by AV, 9-Aug-2024.)

Hypotheses

Ref	Expression
opabex3rd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
opabex3rd.2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → {𝑥 ∣ 𝜓} ∈ V)

Assertion

Ref	Expression
opabex3rd	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} ∈ V)

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

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

Proof of Theorem opabex3rd

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

Step	Hyp	Ref	Expression
1		19.42v 1953	. . . . . . 7 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2		an12 645	. . . . . . . 8 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
3	2	exbii 1848	. . . . . . 7 ⊢ (∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∃𝑥(𝑦 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
4		elxp 5677	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥 ∣ 𝜓} ∧ 𝑣 ∈ {𝑦})))
5		ancom 460	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ {𝑥 ∣ 𝜓} ∧ 𝑣 ∈ {𝑦}) ↔ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}))
6	5	anbi2i 623	. . . . . . . . . . 11 ⊢ ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥 ∣ 𝜓} ∧ 𝑣 ∈ {𝑦})) ↔ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
7	6	2exbii 1849	. . . . . . . . . 10 ⊢ (∃𝑤∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥 ∣ 𝜓} ∧ 𝑣 ∈ {𝑦})) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
8	4, 7	bitri 275	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
9		an12 645	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ (𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
10		velsn 4617	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑦} ↔ 𝑣 = 𝑦)
11	10	anbi1i 624	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ (𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
12	9, 11	bitri 275	. . . . . . . . . . . 12 ⊢ ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ (𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
13	12	exbii 1848	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ ∃𝑣(𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
14		opeq2 4850	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑦⟩)
15	14	eqeq2d 2746	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑦 → (𝑧 = ⟨𝑤, 𝑣⟩ ↔ 𝑧 = ⟨𝑤, 𝑦⟩))
16	15	anbi1d 631	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑦 → ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
17	16	equsexvw 2004	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}))
18	13, 17	bitri 275	. . . . . . . . . 10 ⊢ (∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}))
19	18	exbii 1848	. . . . . . . . 9 ⊢ (∃𝑤∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ ∃𝑤(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}))
20		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑧 = ⟨𝑤, 𝑦⟩
21		nfsab1 2721	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 ∈ {𝑥 ∣ 𝜓}
22	20, 21	nfan 1899	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})
23		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
24		opeq1 4849	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → ⟨𝑤, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
25	24	eqeq2d 2746	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
26		df-clab 2714	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑥 ∣ 𝜓} ↔ [𝑤 / 𝑥]𝜓)
27		sbequ12 2251	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
28	27	equcoms 2019	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
29	26, 28	bitr4id 290	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓))
30	25, 29	anbi12d 632	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → ((𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
31	22, 23, 30	cbvexv1 2343	. . . . . . . . 9 ⊢ (∃𝑤(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
32	8, 19, 31	3bitri 297	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
33	32	anbi2i 623	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
34	1, 3, 33	3bitr4ri 304	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)))
35	34	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})) ↔ ∃𝑦∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)))
36		excom 2162	. . . . 5 ⊢ (∃𝑦∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)))
37	35, 36	bitri 275	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)))
38		eliun 4971	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}))
39		df-rex 3061	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})))
40	38, 39	bitri 275	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})))
41		elopab 5502	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)))
42	37, 40, 41	3bitr4i 303	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)})
43	42	eqriv 2732	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) = {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)}
44		opabex3rd.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
45		opabex3rd.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → {𝑥 ∣ 𝜓} ∈ V)
46		vsnex 5404	. . . . 5 ⊢ {𝑦} ∈ V
47		xpexg 7742	. . . . 5 ⊢ (({𝑥 ∣ 𝜓} ∈ V ∧ {𝑦} ∈ V) → ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
48	45, 46, 47	sylancl 586	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
49	48	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
50		iunexg 7960	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
51	44, 49, 50	syl2anc 584	. 2 ⊢ (𝜑 → ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
52	43, 51	eqeltrrid 2839	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779  [wsb 2064   ∈ wcel 2108  {cab 2713  ∀wral 3051  ∃wrex 3060  Vcvv 3459  {csn 4601  ⟨cop 4607  ∪ ciun 4967  {copab 5181   × cxp 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-opab 5182  df-xp 5660  df-rel 5661

This theorem is referenced by:  satfvsuclem1  35327  satf0suclem  35343  fmlasuc0  35352