Theorem relopabVD 45710

Description: Virtual deduction proof of relopab 5809. 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 5809 is relopabVD 45710 without virtual deductions and was automatically derived from relopabVD 45710.

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 3457	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3457	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 476	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5533	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6	3	biantru 539	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	6	exbii 1881	. . . . . . . . 9 ⊢ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
8	7	exbii 1881	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
9	8	abbii 2829	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
10		ax6ev 2002	. . . . . . . . . . . . . . 15 ⊢ ∃𝑢 𝑢 = 𝑥
11		equcom 2051	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 ↔ 𝑥 = 𝑢)
12	11	exbii 1881	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 𝑢 = 𝑥 ↔ ∃𝑢 𝑥 = 𝑢)
13	10, 12	mpbi 233	. . . . . . . . . . . . . 14 ⊢ ∃𝑢 𝑥 = 𝑢
14		ax6ev 2002	. . . . . . . . . . . . . . . . . . 19 ⊢ ∃𝑣 𝑣 = 𝑦
15		equcom 2051	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑦 ↔ 𝑦 = 𝑣)
16	15	exbii 1881	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑣 𝑣 = 𝑦 ↔ ∃𝑣 𝑦 = 𝑣)
17	14, 16	mpbi 233	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑣 𝑦 = 𝑣
18		idn1 45384	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
19		opeq2 4837	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
20	18, 19	e1a 45437	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (   𝑦 = 𝑣   ▶   ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩   )
21		idn2 45423	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
22		opeq1 4836	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
23	21, 22	e2 45441	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩   )
24		eqeq1 2766	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩))
25	24	biimprd 251	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩))
26	20, 23, 25	e12 45533	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩   )
27		eqeq2 2774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑢, 𝑣⟩))
28	27	biimpd 232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
29	26, 28	e2 45441	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
30	29	in2 45415	. . . . . . . . . . . . . . . . . . . 20 ⊢ (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
31	30	in1 45381	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
32	31	eximi 1868	. . . . . . . . . . . . . . . . . 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 1981	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
35		19.37v 2030	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
36	35	biimpi 219	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
37	34, 36	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
38	37	eximi 1868	. . . . . . . . . . . . . 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 1981	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
41	40	eximi 1868	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
42		19.9v 2017	. . . . . . . . . . . 12 ⊢ (∃𝑦∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
43	42	biimpi 219	. . . . . . . . . . 11 ⊢ (∃𝑦∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
44	41, 43	syl 18	. . . . . . . . . 10 ⊢ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
45	44	eximi 1868	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
46		19.9v 2017	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
47	46	biimpi 219	. . . . . . . . 9 ⊢ (∃𝑥∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
48	45, 47	syl 18	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
49	48	ss2abi 4017	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
50	9, 49	eqsstrri 3981	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
51		vex 3457	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
52		vex 3457	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
53	51, 52	pm3.2i 476	. . . . . . . . . 10 ⊢ (𝑢 ∈ V ∧ 𝑣 ∈ V)
54	53	biantru 539	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ ↔ (𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
55	54	exbii 1881	. . . . . . . 8 ⊢ (∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
56	55	exbii 1881	. . . . . . 7 ⊢ (∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
57	56	abbii 2829	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
58	50, 57	sseqtri 3982	. . . . 5 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
59		df-opab 5172	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
60		df-opab 5172	. . . . 5 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
61	58, 59, 60	3sstr4i 3985	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
62		df-xp 5665	. . . . 5 ⊢ (V × V) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
63	62	eqcomi 2771	. . . 4 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = (V × V)
64	61, 63	sseqtri 3982	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ (V × V)
65	5, 64	sstri 3943	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
66		df-rel 5666	. . 3 ⊢ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V))
67	66	biimpri 231	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V) → Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑})
68	65, 67	e0a 45581	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740  Vcvv 3453   ⊆ wss 3902  ⟨cop 4593  {copab 5171   × cxp 5657  Rel wrel 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5172  df-xp 5665  df-rel 5666  df-vd1 45380  df-vd2 45388

This theorem is used by: (None)