Theorem relopabVD 44894

Description: Virtual deduction proof of relopab 5767. 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 5767 is relopabVD 44894 without virtual deductions and was automatically derived from relopabVD 44894.

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 3440	. . . . . 6 ⊢ 𝑥 ∈ V
2		vex 3440	. . . . . 6 ⊢ 𝑦 ∈ V
3	1, 2	pm3.2i 470	. . . . 5 ⊢ (𝑥 ∈ V ∧ 𝑦 ∈ V)
4	3	a1i 11	. . . 4 ⊢ (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
5	4	ssopab2i 5493	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
6	3	biantru 529	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
7	6	exbii 1848	. . . . . . . . 9 ⊢ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
8	7	exbii 1848	. . . . . . . 8 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
9	8	abbii 2796	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
10		ax6ev 1969	. . . . . . . . . . . . . . 15 ⊢ ∃𝑢 𝑢 = 𝑥
11		equcom 2018	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = 𝑥 ↔ 𝑥 = 𝑢)
12	11	exbii 1848	. . . . . . . . . . . . . . 15 ⊢ (∃𝑢 𝑢 = 𝑥 ↔ ∃𝑢 𝑥 = 𝑢)
13	10, 12	mpbi 230	. . . . . . . . . . . . . 14 ⊢ ∃𝑢 𝑥 = 𝑢
14		ax6ev 1969	. . . . . . . . . . . . . . . . . . 19 ⊢ ∃𝑣 𝑣 = 𝑦
15		equcom 2018	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = 𝑦 ↔ 𝑦 = 𝑣)
16	15	exbii 1848	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃𝑣 𝑣 = 𝑦 ↔ ∃𝑣 𝑦 = 𝑣)
17	14, 16	mpbi 230	. . . . . . . . . . . . . . . . . 18 ⊢ ∃𝑣 𝑦 = 𝑣
18		idn1 44568	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
19		opeq2 4825	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
20	18, 19	e1a 44621	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (   𝑦 = 𝑣   ▶   ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩   )
21		idn2 44607	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
22		opeq1 4824	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
23	21, 22	e2 44625	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩   )
24		eqeq1 2733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩))
25	24	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩))
26	20, 23, 25	e12 44717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩   )
27		eqeq2 2741	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑢, 𝑣⟩))
28	27	biimpd 229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
29	26, 28	e2 44625	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
30	29	in2 44599	. . . . . . . . . . . . . . . . . . . 20 ⊢ (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
31	30	in1 44565	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
32	31	eximi 1835	. . . . . . . . . . . . . . . . . 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 1948	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
35		19.37v 1997	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
36	35	biimpi 216	. . . . . . . . . . . . . . . 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 1835	. . . . . . . . . . . . . 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 1948	. . . . . . . . . . . 12 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
41	40	eximi 1835	. . . . . . . . . . 11 ⊢ (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
42		19.9v 1984	. . . . . . . . . . . 12 ⊢ (∃𝑦∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
43	42	biimpi 216	. . . . . . . . . . 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 1835	. . . . . . . . 9 ⊢ (∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
46		19.9v 1984	. . . . . . . . . 10 ⊢ (∃𝑥∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
47	46	biimpi 216	. . . . . . . . 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 4019	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑥∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
50	9, 49	eqsstrri 3983	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
51		vex 3440	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
52		vex 3440	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
53	51, 52	pm3.2i 470	. . . . . . . . . 10 ⊢ (𝑢 ∈ V ∧ 𝑣 ∈ V)
54	53	biantru 529	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑢, 𝑣⟩ ↔ (𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
55	54	exbii 1848	. . . . . . . 8 ⊢ (∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
56	55	exbii 1848	. . . . . . 7 ⊢ (∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
57	56	abbii 2796	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑢∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
58	50, 57	sseqtri 3984	. . . . 5 ⊢ {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
59		df-opab 5155	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
60		df-opab 5155	. . . . 5 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = {𝑧 ∣ ∃𝑢∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
61	58, 59, 60	3sstr4i 3987	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
62		df-xp 5625	. . . . 5 ⊢ (V × V) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
63	62	eqcomi 2738	. . . 4 ⊢ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = (V × V)
64	61, 63	sseqtri 3984	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ (V × V)
65	5, 64	sstri 3945	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
66		df-rel 5626	. . 3 ⊢ (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V))
67	66	biimpri 228	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V) → Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑})
68	65, 67	e0a 44765	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  Vcvv 3436   ⊆ wss 3903  ⟨cop 4583  {copab 5154   × cxp 5617  Rel wrel 5624

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5155  df-xp 5625  df-rel 5626  df-vd1 44564  df-vd2 44572

This theorem is referenced by: (None)