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Theorem relopabVD 38620
Description: Virtual deduction proof of relopab 5207. 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 5207 is relopabVD 38620 without virtual deductions and was automatically derived from relopabVD 38620.
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 3189 . . . . . 6 𝑥 ∈ V
2 vex 3189 . . . . . 6 𝑦 ∈ V
31, 2pm3.2i 471 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . . 4 (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 4963 . . 3 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)}
63biantru 526 . . . . . . . . . 10 (𝑧 = ⟨𝑥, 𝑦⟩ ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
76exbii 1771 . . . . . . . . 9 (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
87exbii 1771 . . . . . . . 8 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)))
98abbii 2736 . . . . . . 7 {𝑧 ∣ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
10 ax6ev 1887 . . . . . . . . . . . . . . 15 𝑢 𝑢 = 𝑥
11 equcom 1942 . . . . . . . . . . . . . . . 16 (𝑢 = 𝑥𝑥 = 𝑢)
1211exbii 1771 . . . . . . . . . . . . . . 15 (∃𝑢 𝑢 = 𝑥 ↔ ∃𝑢 𝑥 = 𝑢)
1310, 12mpbi 220 . . . . . . . . . . . . . 14 𝑢 𝑥 = 𝑢
14 ax6ev 1887 . . . . . . . . . . . . . . . . . . 19 𝑣 𝑣 = 𝑦
15 equcom 1942 . . . . . . . . . . . . . . . . . . . 20 (𝑣 = 𝑦𝑦 = 𝑣)
1615exbii 1771 . . . . . . . . . . . . . . . . . . 19 (∃𝑣 𝑣 = 𝑦 ↔ ∃𝑣 𝑦 = 𝑣)
1714, 16mpbi 220 . . . . . . . . . . . . . . . . . 18 𝑣 𝑦 = 𝑣
18 idn1 38272 . . . . . . . . . . . . . . . . . . . . . . . 24 (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
19 opeq2 4371 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
2018, 19e1a 38334 . . . . . . . . . . . . . . . . . . . . . . 23 (   𝑦 = 𝑣   ▶   𝑥, 𝑦⟩ = ⟨𝑥, 𝑣   )
21 idn2 38320 . . . . . . . . . . . . . . . . . . . . . . . 24 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
22 opeq1 4370 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
2321, 22e2 38338 . . . . . . . . . . . . . . . . . . . . . . 23 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥, 𝑣⟩ = ⟨𝑢, 𝑣   )
24 eqeq1 2625 . . . . . . . . . . . . . . . . . . . . . . . 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 38433 . . . . . . . . . . . . . . . . . . . . . 22 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥, 𝑦⟩ = ⟨𝑢, 𝑣   )
27 eqeq2 2632 . . . . . . . . . . . . . . . . . . . . . . 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 38338 . . . . . . . . . . . . . . . . . . . . 21 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
3029in2 38312 . . . . . . . . . . . . . . . . . . . 20 (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
3130in1 38269 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑣 → (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)))
3231eximi 1759 . . . . . . . . . . . . . . . . . 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 1908 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑢 → ∃𝑣(𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))
35 19.37v 1907 . . . . . . . . . . . . . . . . 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 1759 . . . . . . . . . . . . . 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 1908 . . . . . . . . . . . 12 (𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
4140eximi 1759 . . . . . . . . . . 11 (∃𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑦𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
42 19.9v 1893 . . . . . . . . . . . 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 1759 . . . . . . . . 9 (∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩ → ∃𝑥𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩)
46 19.9v 1893 . . . . . . . . . 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 3653 . . . . . . 7 {𝑧 ∣ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
509, 49eqsstr3i 3615 . . . . . 6 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
51 vex 3189 . . . . . . . . . . 11 𝑢 ∈ V
52 vex 3189 . . . . . . . . . . 11 𝑣 ∈ V
5351, 52pm3.2i 471 . . . . . . . . . 10 (𝑢 ∈ V ∧ 𝑣 ∈ V)
5453biantru 526 . . . . . . . . 9 (𝑧 = ⟨𝑢, 𝑣⟩ ↔ (𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5554exbii 1771 . . . . . . . 8 (∃𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5655exbii 1771 . . . . . . 7 (∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩ ↔ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V)))
5756abbii 2736 . . . . . 6 {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩} = {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
5850, 57sseqtri 3616 . . . . 5 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
59 df-opab 4674 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
60 df-opab 4674 . . . . 5 {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
6158, 59, 603sstr4i 3623 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
62 df-xp 5080 . . . . 5 (V × V) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
6362eqcomi 2630 . . . 4 {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = (V × V)
6461, 63sseqtri 3616 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ (V × V)
655, 64sstri 3592 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
66 df-rel 5081 . . 3 (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V))
6766biimpri 218 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V) → Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6865, 67e0a 38481 1 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  Vcvv 3186  wss 3555  cop 4154  {copab 4672   × cxp 5072  Rel wrel 5079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-opab 4674  df-xp 5080  df-rel 5081  df-vd1 38268  df-vd2 38276
This theorem is referenced by: (None)
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