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Theorem relopabVD 38659
Description: Virtual deduction proof of relopab 5217. 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 5217 is relopabVD 38659 without virtual deductions and was automatically derived from relopabVD 38659.
 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.
StepHypRef Expression
1 vex 3193 . . . . . 6 𝑥 ∈ V
2 vex 3193 . . . . . 6 𝑦 ∈ V
31, 2pm3.2i 471 . . . . 5 (𝑥 ∈ V ∧ 𝑦 ∈ V)
43a1i 11 . . . 4 (𝜑 → (𝑥 ∈ V ∧ 𝑦 ∈ V))
54ssopab2i 4973 . . 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 38311 . . . . . . . . . . . . . . . . . . . . . . . 24 (   𝑦 = 𝑣   ▶   𝑦 = 𝑣   )
19 opeq2 4378 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = 𝑣 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑣⟩)
2018, 19e1a 38373 . . . . . . . . . . . . . . . . . . . . . . 23 (   𝑦 = 𝑣   ▶   𝑥, 𝑦⟩ = ⟨𝑥, 𝑣   )
21 idn2 38359 . . . . . . . . . . . . . . . . . . . . . . . 24 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   𝑥 = 𝑢   )
22 opeq1 4377 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑢 → ⟨𝑥, 𝑣⟩ = ⟨𝑢, 𝑣⟩)
2321, 22e2 38377 . . . . . . . . . . . . . . . . . . . . . . 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 38472 . . . . . . . . . . . . . . . . . . . . . 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 38377 . . . . . . . . . . . . . . . . . . . . 21 (   𝑦 = 𝑣   ,   𝑥 = 𝑢   ▶   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩)   )
3029in2 38351 . . . . . . . . . . . . . . . . . . . 20 (   𝑦 = 𝑣   ▶   (𝑥 = 𝑢 → (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑧 = ⟨𝑢, 𝑣⟩))   )
3130in1 38308 . . . . . . . . . . . . . . . . . . 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 3659 . . . . . . 7 {𝑧 ∣ ∃𝑥𝑦 𝑧 = ⟨𝑥, 𝑦⟩} ⊆ {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
509, 49eqsstr3i 3621 . . . . . 6 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣 𝑧 = ⟨𝑢, 𝑣⟩}
51 vex 3193 . . . . . . . . . . 11 𝑢 ∈ V
52 vex 3193 . . . . . . . . . . 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 3622 . . . . 5 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))} ⊆ {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
59 df-opab 4684 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V))}
60 df-opab 4684 . . . . 5 {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = {𝑧 ∣ ∃𝑢𝑣(𝑧 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ V ∧ 𝑣 ∈ V))}
6158, 59, 603sstr4i 3629 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
62 df-xp 5090 . . . . 5 (V × V) = {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)}
6362eqcomi 2630 . . . 4 {⟨𝑢, 𝑣⟩ ∣ (𝑢 ∈ V ∧ 𝑣 ∈ V)} = (V × V)
6461, 63sseqtri 3622 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ V ∧ 𝑦 ∈ V)} ⊆ (V × V)
655, 64sstri 3597 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V)
66 df-rel 5091 . . 3 (Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V))
6766biimpri 218 . 2 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ (V × V) → Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6865, 67e0a 38520 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 3190   ⊆ wss 3560  ⟨cop 4161  {copab 4682   × cxp 5082  Rel wrel 5089 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 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-opab 4684  df-xp 5090  df-rel 5091  df-vd1 38307  df-vd2 38315 This theorem is referenced by: (None)
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