Theorem opabex3rd 7667

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

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)

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 1954	. . . . . . 7 ⊢ (∃𝑥(𝑦 ∈ 𝐴 ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
2		an12 643	. . . . . . . 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 5578	. . . . . . . . . 10 ⊢ (𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ {𝑥 ∣ 𝜓} ∧ 𝑣 ∈ {𝑦})))
5		ancom 463	. . . . . . . . . . . 12 ⊢ ((𝑤 ∈ {𝑥 ∣ 𝜓} ∧ 𝑣 ∈ {𝑦}) ↔ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}))
6	5	anbi2i 624	. . . . . . . . . . 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 277	. . . . . . . . 9 ⊢ (𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑤∃𝑣(𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
9		an12 643	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑤, 𝑣⟩ ∧ (𝑣 ∈ {𝑦} ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ (𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
10		velsn 4583	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ {𝑦} ↔ 𝑣 = 𝑦)
11	10	anbi1i 625	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ {𝑦} ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ (𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})))
12	9, 11	bitri 277	. . . . . . . . . . . 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 4804	. . . . . . . . . . . . . 14 ⊢ (𝑣 = 𝑦 → ⟨𝑤, 𝑣⟩ = ⟨𝑤, 𝑦⟩)
15	14	eqeq2d 2832	. . . . . . . . . . . . 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 2011	. . . . . . . . . . 11 ⊢ (∃𝑣(𝑣 = 𝑦 ∧ (𝑧 = ⟨𝑤, 𝑣⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})) ↔ (𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}))
18	13, 17	bitri 277	. . . . . . . . . 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 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑧 = ⟨𝑤, 𝑦⟩
21		nfsab1 2808	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 ∈ {𝑥 ∣ 𝜓}
22	20, 21	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓})
23		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑤(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)
24		opeq1 4803	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → ⟨𝑤, 𝑦⟩ = ⟨𝑥, 𝑦⟩)
25	24	eqeq2d 2832	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑧 = ⟨𝑤, 𝑦⟩ ↔ 𝑧 = ⟨𝑥, 𝑦⟩))
26		sbequ12 2253	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
27	26	equcoms 2027	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝜓 ↔ [𝑤 / 𝑥]𝜓))
28		df-clab 2800	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑥 ∣ 𝜓} ↔ [𝑤 / 𝑥]𝜓)
29	27, 28	syl6rbbr 292	. . . . . . . . . . 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 2362	. . . . . . . . 9 ⊢ (∃𝑤(𝑧 = ⟨𝑤, 𝑦⟩ ∧ 𝑤 ∈ {𝑥 ∣ 𝜓}) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
32	8, 19, 31	3bitri 299	. . . . . . . 8 ⊢ (𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
33	32	anbi2i 624	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)))
34	1, 3, 33	3bitr4ri 306	. . . . . 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 2169	. . . . 5 ⊢ (∃𝑦∃𝑥(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)))
37	35, 36	bitri 277	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)))
38		eliun 4923	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}))
39		df-rex 3144	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})))
40	38, 39	bitri 277	. . . 4 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ({𝑥 ∣ 𝜓} × {𝑦})))
41		elopab 5414	. . . 4 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝜓)))
42	37, 40, 41	3bitr4i 305	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ↔ 𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)})
43	42	eqriv 2818	. 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		snex 5332	. . . . 5 ⊢ {𝑦} ∈ V
47		xpexg 7473	. . . . 5 ⊢ (({𝑥 ∣ 𝜓} ∈ V ∧ {𝑦} ∈ V) → ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
48	45, 46, 47	sylancl 588	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
49	48	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
50		iunexg 7664	. . 3 ⊢ ((𝐴 ∈ V ∧ ∀𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V) → ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
51	44, 49, 50	syl2anc 586	. 2 ⊢ (𝜑 → ∪ 𝑦 ∈ 𝐴 ({𝑥 ∣ 𝜓} × {𝑦}) ∈ V)
52	43, 51	eqeltrrid 2918	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜓)} ∈ V)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780  [wsb 2069   ∈ wcel 2114  {cab 2799  ∀wral 3138  ∃wrex 3139  Vcvv 3494  {csn 4567  ⟨cop 4573  ∪ ciun 4919  {copab 5128   × cxp 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363

This theorem is referenced by:  satfvsuclem1  32606  satf0suclem  32622  fmlasuc0  32631