Theorem relopabVD 42831

Description: Virtual deduction proof of relopab 5760. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. relopab 5760 is relopabVD 42831 without virtual deductions and was automatically derived from relopabVD 42831.

1::	⊢ (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
2:1:	⊢ (   𝑦 = 𝑣   ▶   ⟨𝑥   ,   𝑦⟩ = ⟨𝑥   ,   𝑣 ⟩   )
3::	⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
4:3:	⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥   ,   𝑣⟩ = ⟨ 𝑢, 𝑣⟩   )
5:2,4:	⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥   ,   𝑦⟩ = ⟨ 𝑢, 𝑣⟩   )
6:5:	⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥   ,   𝑦 ⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
7:6:	⊢ (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,    𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
8:7:	⊢ (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦 ⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
9:8:	⊢ (∃𝑣𝑦 = 𝑣 → ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
90::	⊢ (𝑣 = 𝑦 ↔ 𝑦 = 𝑣)
91:90:	⊢ (∃𝑣𝑣 = 𝑦 ↔ ∃𝑣𝑦 = 𝑣)
92::	⊢ ∃𝑣𝑣 = 𝑦
10:91,92:	⊢ ∃𝑣𝑦 = 𝑣
11:9,10:	⊢ ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
12:11:	⊢ (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
13::	⊢ (∃𝑣(𝑧 = ⟨𝑥   ,   𝑦⟩ → 𝑧 = ⟨𝑢 , 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩))
14:12,13:	⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
15:14:	⊢ (∃𝑢𝑥 = 𝑢 → ∃𝑢(𝑧 = ⟨𝑥   ,   𝑦 ⟩ → ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩))
150::	⊢ (𝑢 = 𝑥 ↔ 𝑥 = 𝑢)
151:150:	⊢ (∃𝑢𝑢 = 𝑥 ↔ ∃𝑢𝑥 = 𝑢)
152::	⊢ ∃𝑢𝑢 = 𝑥
16:151,152:	⊢ ∃𝑢𝑥 = 𝑢
17:15,16:	⊢ ∃𝑢(𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑣𝑧 = ⟨ 𝑢, 𝑣⟩)
18:17:	⊢ (𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢∃𝑣𝑧 = ⟨ 𝑢, 𝑣⟩)
19:18:	⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑦∃𝑢 ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
20::	⊢ (∃𝑦∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
21:19,20:	⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
22:21:	⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑥 ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
23::	⊢ (∃𝑥∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
24:22,23:	⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢 ∃𝑣𝑧 = ⟨𝑢, 𝑣⟩)
25:24:	⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩}
26::	⊢ 𝑥 ∈ V
27::	⊢ 𝑦 ∈ V
28:26,27:	⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
29:28:	⊢ (𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ (𝑧 = ⟨𝑥   ,   𝑦 ⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
30:29:	⊢ (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
31:30:	⊢ (∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑥 ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
32:31:	⊢ {𝑧 ∣ ∃𝑥∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} = { 𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
320:25,32:	⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢, 𝑣⟩}
33::	⊢ 𝑢 ∈ V
34::	⊢ 𝑣 ∈ V
35:33,34:	⊢ (𝑢 ∈ V ∧ 𝑣 ∈ V)
36:35:	⊢ (𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ (𝑧 = ⟨𝑢   ,   𝑣 ⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
37:36:	⊢ (∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
38:37:	⊢ (∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑢 ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
39:38:	⊢ {𝑧 ∣ ∃𝑢∃𝑣𝑧 = ⟨𝑢   ,   𝑣⟩} = { 𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
40:320,39:	⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
41::	⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) }
42::	⊢ {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)) }
43:40,41,42:	⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
44::	⊢ {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = (V × V)
45:43,44:	⊢ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} ⊆ (V × V)
46:28:	⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
47:46:	⊢ {⟨𝑥   ,   𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
48:45,47:	⊢ {⟨𝑥   ,   𝑦⟩ ∣ 𝜑} ⊆ (V × V)
qed:48:	⊢ Rel {⟨𝑥   ,   𝑦⟩ ∣ 𝜑}

(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
relopabVD	⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem relopabVD

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

Step	Hyp	Ref	Expression
1		vex 3445	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3445	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 471	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5488	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6	3	biantru 530	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	6	exbii 1849	. . . . . . . . 9 ⊢ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
8	7	exbii 1849	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
9	8	abbii 2806	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
10		ax6ev 1972	. . . . . . . . . . . . . . 15 ⊢ ∃𝑢 𝑢 = 𝑥
11		equcom 2020	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 ↔ 𝑥 = 𝑢)
12	11	exbii 1849	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 𝑢 = 𝑥 ↔ ∃𝑢 𝑥 = 𝑢)
13	10, 12	mpbi 229	. . . . . . . . . . . . . 14 ⊢ ∃𝑢 𝑥 = 𝑢
14		ax6ev 1972	. . . . . . . . . . . . . . . . . . 19 ⊢ ∃𝑣 𝑣 = 𝑦
15		equcom 2020	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑦 ↔ 𝑦 = 𝑣)
16	15	exbii 1849	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑣 𝑣 = 𝑦 ↔ ∃𝑣 𝑦 = 𝑣)
17	14, 16	mpbi 229	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑣 𝑦 = 𝑣
18		idn1 42504	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
19		opeq2 4817	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
20	18, 19	e1a 42557	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (   𝑦 = 𝑣   ▶   ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩   )
21		idn2 42543	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
22		opeq1 4816	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
23	21, 22	e2 42561	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩   )
24		eqeq1 2740	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩))
25	24	biimprd 247	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩))
26	20, 23, 25	e12 42654	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩   )
27		eqeq2 2748	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑢, 𝑣⟩))
28	27	biimpd 228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
29	26, 28	e2 42561	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
30	29	in2 42535	. . . . . . . . . . . . . . . . . . . 20 ⊢ (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
31	30	in1 42501	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
32	31	eximi 1836	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑣 𝑦 = 𝑣 → ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
33	17, 32	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
34	33	19.37iv 1951	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
35		19.37v 1994	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
36	35	biimpi 215	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
37	34, 36	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
38	37	eximi 1836	. . . . . . . . . . . . . 14 ⊢ (∃𝑢 𝑥 = 𝑢 → ∃𝑢(𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
39	13, 38	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ∃𝑢(𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
40	39	19.37iv 1951	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
41	40	eximi 1836	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
42		19.9v 1986	. . . . . . . . . . . 12 ⊢ (∃𝑦∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
43	42	biimpi 215	. . . . . . . . . . 11 ⊢ (∃𝑦∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
44	41, 43	syl 17	. . . . . . . . . 10 ⊢ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
45	44	eximi 1836	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
46		19.9v 1986	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
47	46	biimpi 215	. . . . . . . . 9 ⊢ (∃𝑥∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
48	45, 47	syl 17	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
49	48	ss2abi 4010	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
50	9, 49	eqsstrri 3966	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
51		vex 3445	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
52		vex 3445	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
53	51, 52	pm3.2i 471	. . . . . . . . . 10 ⊢ (𝑢 ∈ V ∧ 𝑣 ∈ V)
54	53	biantru 530	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ ↔ (𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
55	54	exbii 1849	. . . . . . . 8 ⊢ (∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
56	55	exbii 1849	. . . . . . 7 ⊢ (∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
57	56	abbii 2806	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
58	50, 57	sseqtri 3967	. . . . 5 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
59		df-opab 5152	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
60		df-opab 5152	. . . . 5 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
61	58, 59, 60	3sstr4i 3974	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
62		df-xp 5620	. . . . 5 ⊢ (V × V) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
63	62	eqcomi 2745	. . . 4 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = (V × V)
64	61, 63	sseqtri 3967	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ (V × V)
65	5, 64	sstri 3940	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
66		df-rel 5621	. . 3 ⊢ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V))
67	66	biimpri 227	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V) → Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑})
68	65, 67	e0a 42702	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540  ∃wex 1780   ∈ wcel 2105  {cab 2713  Vcvv 3441   ⊆ wss 3897  ⟨cop 4578  {copab 5151   × cxp 5612  Rel wrel 5619

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-opab 5152  df-xp 5620  df-rel 5621  df-vd1 42500  df-vd2 42508

This theorem is referenced by: (None)