Theorem oprabv 7487

Description: If a pair and a class are in a relationship given by a class abstraction of a collection of nested ordered pairs, the involved classes are sets. (Contributed by Alexander van der Vekens, 8-Jul-2018.)

Assertion

Ref	Expression
oprabv	⊢ (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))

Distinct variable groups:   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧   𝑥,𝑍,𝑦,𝑧

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

Proof of Theorem oprabv

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reloprab 7486	. . 3 ⊢ Rel {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
2	1	brrelex12i 5709	. 2 ⊢ (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V))
3		df-br 5106	. . . . 5 ⊢ (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 ↔ ⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
4		opex 5435	. . . . . . . . 9 ⊢ ⟨𝑋, 𝑌⟩ ∈ V
5		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤⟨𝑋, 𝑌⟩
6	5	nfeq1 2940	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩
7		nfv 1948	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤𝜑
8	6, 7	nfan 1933	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
9	8	nfex 2357	. . . . . . . . . . 11 ⊢ Ⅎ𝑤∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
10	9	nfex 2357	. . . . . . . . . 10 ⊢ Ⅎ𝑤∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)
11		nfcv 2925	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧⟨𝑋, 𝑌⟩
12	11	nfeq1 2940	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩
13		nfsbc1v 3761	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧[𝑍 / 𝑧]𝜑
14	12, 13	nfan 1933	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)
15	14	nfex 2357	. . . . . . . . . . 11 ⊢ Ⅎ𝑧∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)
16	15	nfex 2357	. . . . . . . . . 10 ⊢ Ⅎ𝑧∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)
17		eqeq1 2767	. . . . . . . . . . . 12 ⊢ (𝑤 = ⟨𝑋, 𝑌⟩ → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩))
18	17	anbi1d 643	. . . . . . . . . . 11 ⊢ (𝑤 = ⟨𝑋, 𝑌⟩ → ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
19	18	2exbidv 1958	. . . . . . . . . 10 ⊢ (𝑤 = ⟨𝑋, 𝑌⟩ → (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑)))
20		sbceq1a 3752	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑍 → (𝜑 ↔ [𝑍 / 𝑧]𝜑))
21	20	anbi2d 642	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑍 → ((⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)))
22	21	2exbidv 1958	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)))
23	10, 16, 19, 22	opelopabgf 5518	. . . . . . . . 9 ⊢ ((⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V) → (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)))
24	4, 23	mpan 703	. . . . . . . 8 ⊢ (𝑍 ∈ V → (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} ↔ ∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑)))
25		eqcom 2770	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
26		vex 3457	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ V
27		vex 3457	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
28	26, 27	opth 5448	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩ ↔ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
29	25, 28	bitri 278	. . . . . . . . . . . . . 14 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌))
30		eqvisset 3473	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → 𝑋 ∈ V)
31		eqvisset 3473	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑌 → 𝑌 ∈ V)
32	30, 31	anim12i 625	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
33	29, 32	sylbi 220	. . . . . . . . . . . . 13 ⊢ (⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ → (𝑋 ∈ V ∧ 𝑌 ∈ V))
34	33	adantr 486	. . . . . . . . . . . 12 ⊢ ((⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
35	34	exlimivv 1966	. . . . . . . . . . 11 ⊢ (∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V))
36	35	anim1i 627	. . . . . . . . . 10 ⊢ ((∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) ∧ 𝑍 ∈ V) → ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ V))
37		df-3an 1105	. . . . . . . . . 10 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V) ↔ ((𝑋 ∈ V ∧ 𝑌 ∈ V) ∧ 𝑍 ∈ V))
38	36, 37	sylibr 237	. . . . . . . . 9 ⊢ ((∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) ∧ 𝑍 ∈ V) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))
39	38	expcom 419	. . . . . . . 8 ⊢ (𝑍 ∈ V → (∃𝑥∃𝑦(⟨𝑋, 𝑌⟩ = ⟨𝑥, 𝑦⟩ ∧ [𝑍 / 𝑧]𝜑) → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
40	24, 39	sylbid 243	. . . . . . 7 ⊢ (𝑍 ∈ V → (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
41	40	com12 33	. . . . . 6 ⊢ (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
42		dfoprab2 7485	. . . . . 6 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
43	41, 42	eleq2s 2881	. . . . 5 ⊢ (⟨⟨𝑋, 𝑌⟩, 𝑍⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
44	3, 43	sylbi 220	. . . 4 ⊢ (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑍 ∈ V → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
45	44	com12 33	. . 3 ⊢ (𝑍 ∈ V → (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
46	45	adantl 487	. 2 ⊢ ((⟨𝑋, 𝑌⟩ ∈ V ∧ 𝑍 ∈ V) → (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V)))
47	2, 46	mpcom 39	1 ⊢ (⟨𝑋, 𝑌⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑍 → (𝑋 ∈ V ∧ 𝑌 ∈ V ∧ 𝑍 ∈ V))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2146  Vcvv 3453  [wsbc 3741  ⟨cop 4837   class class class wbr 5105  {copab 5169  {coprab 7428

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-oprab 7431

This theorem is used by: (None)