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Theorem relopabVD 39652
Description: Virtual deduction proof of relopab 5384. 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 5384 is relopabVD 39652 without virtual deductions and was automatically derived from relopabVD 39652.
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 = 𝑦𝑦 = 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 = 𝑥𝑥 = 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⟩ → ∃𝑢𝑣𝑧 = ⟨ 𝑢, 𝑣⟩)
19:18: (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑦𝑢 𝑣𝑧 = ⟨𝑢, 𝑣⟩)
20:: (∃𝑦𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
21:19,20: (∃𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
22:21: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑥 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
23:: (∃𝑥𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ → 𝑢𝑣𝑧 = ⟨𝑢, 𝑣⟩)
24:22,23: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ → ∃𝑢 𝑣𝑧 = ⟨𝑢, 𝑣⟩)
25:24: {𝑧 ∣ ∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨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: (∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩ ↔ ∃𝑥 𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
32:31: {𝑧 ∣ ∃𝑥𝑦𝑧 = ⟨𝑥   ,   𝑦⟩} = { 𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
320:25,32: {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨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: (∃𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩ ↔ ∃𝑢 𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
39:38: {𝑧 ∣ ∃𝑢𝑣𝑧 = ⟨𝑢   ,   𝑣⟩} = { 𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
40:320,39: {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥   ,   𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
41:: {⟨𝑥   ,   𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V )} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) }
42:: {⟨𝑢   ,   𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V )} = {𝑧 ∣ ∃𝑢𝑣(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.
StepHypRef Expression
1 vex 3354 . . . . . 6 𝑥 ∈ V
2 vex 3354 . . . . . 6 𝑦 ∈ V
31, 2pm3.2i 456 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . . 4 (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 5136 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
63biantru 519 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
76exbii 1924 . . . . . . . . 9 (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
87exbii 1924 . . . . . . . 8 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
98abbii 2888 . . . . . . 7 {𝑧 ∣ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
10 ax6ev 2059 . . . . . . . . . . . . . . 15 𝑢 𝑢 = 𝑥
11 equcom 2103 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥𝑥 = 𝑢)
1211exbii 1924 . . . . . . . . . . . . . . 15 (∃𝑢 𝑢 = 𝑥 ↔ ∃𝑢 𝑥 = 𝑢)
1310, 12mpbi 220 . . . . . . . . . . . . . 14 𝑢 𝑥 = 𝑢
14 ax6ev 2059 . . . . . . . . . . . . . . . . . . 19 𝑣 𝑣 = 𝑦
15 equcom 2103 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑦𝑦 = 𝑣)
1615exbii 1924 . . . . . . . . . . . . . . . . . . 19 (∃𝑣 𝑣 = 𝑦 ↔ ∃𝑣 𝑦 = 𝑣)
1714, 16mpbi 220 . . . . . . . . . . . . . . . . . 18 𝑣 𝑦 = 𝑣
18 idn1 39308 . . . . . . . . . . . . . . . . . . . . . . . 24 (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
19 opeq2 4540 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
2018, 19e1a 39370 . . . . . . . . . . . . . . . . . . . . . . 23 (   𝑦 = 𝑣   ▶   𝑥, 𝑦⟩ = ⟨𝑥, 𝑣   )
21 idn2 39356 . . . . . . . . . . . . . . . . . . . . . . . 24 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
22 opeq1 4539 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
2321, 22e2 39374 . . . . . . . . . . . . . . . . . . . . . . 23 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥, 𝑣⟩ = ⟨𝑢, 𝑣   )
24 eqeq1 2775 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ ↔ ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩))
2524biimprd 238 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩ → (⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩ → ⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩))
2620, 23, 25e12 39469 . . . . . . . . . . . . . . . . . . . . . 22 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥, 𝑦⟩ = ⟨𝑢, 𝑣   )
27 eqeq2 2782 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨𝑢, 𝑣⟩))
2827biimpd 219 . . . . . . . . . . . . . . . . . . . . . 22 (⟨𝑥, 𝑦⟩ = ⟨𝑢, 𝑣⟩ → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
2926, 28e2 39374 . . . . . . . . . . . . . . . . . . . . 21 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
3029in2 39348 . . . . . . . . . . . . . . . . . . . 20 (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
3130in1 39305 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
3231eximi 1910 . . . . . . . . . . . . . . . . . 18 (∃𝑣 𝑦 = 𝑣 → ∃𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
3317, 32ax-mp 5 . . . . . . . . . . . . . . . . 17 𝑣(𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
343319.37iv 2028 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
35 19.37v 2078 . . . . . . . . . . . . . . . . 17 (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
3635biimpi 206 . . . . . . . . . . . . . . . 16 (∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩) → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
3734, 36syl 17 . . . . . . . . . . . . . . 15 (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
3837eximi 1910 . . . . . . . . . . . . . 14 (∃𝑢 𝑥 = 𝑢 → ∃𝑢(𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩))
3913, 38ax-mp 5 . . . . . . . . . . . . 13 𝑢(𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
403919.37iv 2028 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4140eximi 1910 . . . . . . . . . . 11 (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
42 19.9v 2065 . . . . . . . . . . . 12 (∃𝑦𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4342biimpi 206 . . . . . . . . . . 11 (∃𝑦𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4441, 43syl 17 . . . . . . . . . 10 (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4544eximi 1910 . . . . . . . . 9 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
46 19.9v 2065 . . . . . . . . . 10 (∃𝑥𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4746biimpi 206 . . . . . . . . 9 (∃𝑥𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4845, 47syl 17 . . . . . . . 8 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4948ss2abi 3823 . . . . . . 7 {𝑧 ∣ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
509, 49eqsstr3i 3785 . . . . . 6 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
51 vex 3354 . . . . . . . . . . 11 𝑢 ∈ V
52 vex 3354 . . . . . . . . . . 11 𝑣 ∈ V
5351, 52pm3.2i 456 . . . . . . . . . 10 (𝑢 ∈ V ∧ 𝑣 ∈ V)
5453biantru 519 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ ↔ (𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5554exbii 1924 . . . . . . . 8 (∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5655exbii 1924 . . . . . . 7 (∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5756abbii 2888 . . . . . 6 {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩} = {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
5850, 57sseqtri 3786 . . . . 5 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
59 df-opab 4847 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
60 df-opab 4847 . . . . 5 {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
6158, 59, 603sstr4i 3793 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
62 df-xp 5255 . . . . 5 (V × V) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
6362eqcomi 2780 . . . 4 {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = (V × V)
6461, 63sseqtri 3786 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ (V × V)
655, 64sstri 3761 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
66 df-rel 5256 . . 3 (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V))
6766biimpri 218 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V) → Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6865, 67e0a 39517 1 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  {cab 2757  Vcvv 3351  wss 3723  cop 4322  {copab 4846   × cxp 5247  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-opab 4847  df-xp 5255  df-rel 5256  df-vd1 39304  df-vd2 39312
This theorem is referenced by: (None)
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